**1. Introduction**

A special place in mathematical physics is occupied by the problem of exact integration of the equations of motion of a classical or quantum test particle in external electromagnetic and gravitational fields. This problem is closely related to the study of the symmetry of gravitational and electromagnetic fields in which a given particle moves. A necessary condition for the existence of such symmetry is the admissibility of the algebra of symmetry operators, given by vector and tensor Killing fields, for spacetime and the external electromagnetic field. The algebras of these operators are isomorphic to the algebras of the symmetry operators of the equations of motion of a test particle—Hamilton–Jacobi, Klein– Gordon–Fock, or Dirac–Fock. At present, two methods are known for the exact integration of the equations of motion of a test particle. These are the methods of commutative and noncommutative integration. The first method is based on the use of commutative algebra of symmetry operators (integrals of motion) that form a complete set. The complete set includes linear operators of first and second degree in momentum formed by vector and tensor Killing fields of complete sets of geometric objects of *V*4. The method is known as the method of complete separation of variables (in the Hamilton–Jacobi, Klein–Gordon– Fock, or Dirac–Fock equations). The spaces in which the method of complete separation of variables is applicable are called Stackel spaces. The theory of Stackel spaces was developed in [1–12]. A description of the theory and a detailed bibliography can be found in [13–16]. The most frequently used exact solutions of the gravitational field equations in the theory of gravity were constructed on the basis of Stackel spaces (see, e.g., [17–19]). These solutions are still widely used in the study of various effects in gravitational fields (see, e.g., [20–27]).

The second method (noncommutative integration) was developed in [28]. This method is based on the use of algebra of symmetry operators, which are linear in momenta and constructed using Killing vector fields forming noncommutative groups of motion of spacetime *G*3 and *G*4. The algebras of the symmetry operators of the Klein–Gordon– Fock equation constructed using the algebras of the operators of the noncommutative

**Citation:** Obukhov, V.V. Maxwell's Equations in Homogeneous Spaces for Admissible Electromagnetic Fields. *Universe* **2022**, *8*, 245. https:// doi.org/10.3390/universe8040245

Academic Editors: Steven Duplij and Michael L. Walker

Received: 31 March 2022 Accepted: 12 April 2022 Published: 15 April 2022

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**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

motion group of space *V*4 are complemented to a commutative algebra by the operators of differentiation of the first order in 4 essential parameters. Among these spacetime manifolds, the homogeneous spaces are of greatest interest for the theory of gravity (see, e.g., [29–36]).

Thus, these two methods complement each other to a considerable extent and have similar classification problems (by solving the classification problem, we mean enumerating all metrics and electromagnetic potentials that are not equivalent in terms of admissible transformations). Among these classification problems, the most important are the following.

Classification of all metrics of homogeneous and Stackel spaces in privileged coordinate systems. For Stackel spaces, this problem was solved in building the theory of complete separation of variables in the papers cited above. For homogeneous spaces, this problem was solved in the work of Petrov (see [37]).

Classification of all (admissible) electromagnetic fields applicable to these methods. For the Hamilton–Jacobi and Klein–Gordon–Fock equations, this problem is completely solved in homogeneous spaces (see [38–43]). In Stackel spaces, it is completely solved for the Hamilton–Jacobi equation and partially solved for the Klein–Gordon–Fock equation (see [14–16]).

Classification of all vacuum and electrovacuum solutions of the Einstein equations with metrics of Stackel and homogeneous spaces in admissible electromagnetic fields. This problem has been completely solved for the Stackel metric (see [17–20]). However, this classification problem has not ye<sup>t</sup> been studied for homogeneous spaces.

The solutions to these problems can be viewed as stages of the solution of a single classification problem. In the first two stages, we find all relevant gravitational and electromagnetic fields that are not connected by field equations. In the third stage, using the results of the first two stages, we find metrics and electromagnetic potentials that satisfy the Einstein–Maxwell vacuum equations and have physical meaning.

Thus, for the complete solution to the problem of uniform classification, the Einstein– Maxwell vacuum equations must be integrated using the previously found potentials of admissible electromagnetic fields and the known metrics of homogeneous spaces in privileged (canonical) coordinate systems. This problem can also be divided into two stages. In the first stage, all solutions of Maxwell's vacuum equations for the potentials of admissible electromagnetic fields should be found. The present work is devoted to this stage. In the next stage, the plan is to use the obtained results for the integration of the Einstein–Maxwell equations. This will be the subject of further research. The present work is organized as follows.

Section 2 contains information from the theory of homogeneous spaces, which will be used later, and definitions and conditions for the potentials of admissible electromagnetic fields, written in canonical frames associated with motion groups of a homogeneous space.

In the Section 3 Maxwell's vacuum equations are written in canonical frames.

The Section 4 contains all solutions of Maxwell's vacuum equations for homogeneous spaces admitting groups of motions *<sup>G</sup>*3(*I*) − *<sup>G</sup>*3(*<sup>V</sup> I*).

#### **2. Homogeneous Spaces**

By the accepted definition, a spacetime manifold *V*4 is a homogeneous space—if a threeparameter group of motions acts on it—whose transitivity hypersurface *V*3 is endowed with the Euclidean space signature. Let us introduce a semi-geodesic coordinate system [*ui* ], in which the metric *V*4 has the form:

$$ds^2 = \mathcal{g}\_{ij} du^i du^j = -du^{0^2} + \mathcal{g}\_{a\beta} du^a du^\beta, \quad \det|\mathcal{g}\_{a\beta}| > 0. \tag{1}$$

The coordinate indices of the variables of the semi-geodesic coordinate system are denoted by the lower-case Latin letters: *i*, *j*, *k*, *l* = 0, 1, ... , 3. The coordinate indices of the variables of the local coordinate system on the hypersurface *V*3 are denoted by the lowercase Greek letters: *α*, *β*, *γ*, *σ* = 1, ... , 3. A 0 index denotes the temporary variable. Group

indices and indices of nonholonomic frames are denoted by *a*, *d*, *c* = 1, ... , 3. Summation is performed over repeated upper and lower indices within the index range.

There is another (equivalent) definition of a homogeneous space, according to which the spacetime *V*4 is homogeneous if its subspace *V*3, endowed with the Euclidean space signature, admits a set of coordinate transformations (the group *G*3 of motions spaces *V*4) that allow the connection of any two points in *V*3. (see, e.g., [44]). This definition directly implies that the metric tensor of the *V*3 space can be represented as follows:

$$g\_{a\circ} = \epsilon^a\_a \epsilon^b\_{\beta} \eta\_{ab\prime} \quad ||\eta\_{ab}|| = ||a\_{ab}(u^0)||, \quad \epsilon^a\_{a,0} = 0, \quad \det||a\_{ab}|| = l\_0^2,\tag{2}$$
  $\text{form:} \qquad \epsilon^a = \epsilon^a \, du^a$ 

while the

$$
\omega^a = e^a\_a du^a
$$

is invariant under the transformation group *G*3. The vectors of the frame *eaα* (we call them canonical) define a nonholonomic coordinate system in *V*3, and their dual triplet of vectors:

$$e\_{a\prime}^a \quad e\_a^a e\_a^b = \delta\_{a\prime}^b \quad e\_a^a e\_\beta^a = \delta\_\beta^a$$

define the operators of the *G*3 algebra group:

$$
\hat{Y}\_a = \mathcal{e}^a\_a \partial\_{a\prime} \quad [\hat{Y}\_{a\prime} \hat{Y}\_b] = \mathcal{C}^c\_{ab} \hat{Y}\_c.
$$

The Killing vector fields *ξαa* and their dual vector fields *ξaα* form another frame in the space *V*3 (we will call it the Killing frame) and another representation of the algebra of group *G*3. In the dual frame, the metric of the space *V*3 has the form:

$$\mathfrak{g}\_{a\beta} = \mathfrak{J}^{a}\_{a}\mathfrak{J}^{b}\_{\beta}\mathbf{G}\_{ab\prime} \quad \mathfrak{J}^{a}\_{a}\mathfrak{J}^{b}\_{a} = \delta^{b}\_{a\prime} \quad \mathfrak{J}^{a}\_{a}\mathfrak{J}^{a}\_{\beta} = \delta^{a}\_{\beta\prime} \tag{3}$$

where *Gab* are the nonholonomic components of the *gαβ* tensor in this framework. The vector fields *ξαa* satisfy the Killing equations:

$$
\mathcal{g}^{a\beta}\_{\gamma\gamma} \xi^{\gamma}\_a = \mathcal{g}^{a\gamma} \xi^{\beta}\_{a,\gamma} + \mathcal{g}^{\beta\gamma} \xi^a\_{a,\gamma} \tag{4}
$$

and form the infinitesimal group operators of the algebra *G*3:

$$
\hat{X}\_a = \zeta\_a^a \partial\_{a\prime} \quad [\hat{X}\_a, \hat{X}\_b] = \mathcal{C}\_{ab}^c \hat{X}\_c. \tag{5}
$$

The Killing equation in the *ξαa* frame has the following form:

$$\mathcal{G}^{ab}\_{\mid c} = \mathcal{G}^{ad} \mathcal{C}^{b}\_{dc} + \mathcal{G}^{bd} \mathcal{C}^{a}\_{dc} \quad (|a = \mathcal{J}^{a}\_{a} \partial\_{a}). \tag{6}$$

Indeed, substituting the expression:

$$\mathcal{S}^{a\beta} = \mathcal{S}^{a}\_{a}\mathcal{S}^{b}\_{b}G^{ab}$$

into Equation (4), we ge<sup>t</sup>

$$G^{ab} \left( \left( \zeta^a\_{a|c} \mathfrak{J}^{\beta}\_b - \zeta^a\_a \mathfrak{J}^{\beta}\_{c|b} \right) + \left( \zeta^a\_a \mathfrak{J}^{\beta}\_{b|c} - \zeta^\beta\_a \mathfrak{J}^a\_{c|b} \right) \right) + \zeta^a\_a \zeta^\beta\_b G^{ab}\_{|c} = 0.$$

Substituting here the commutation relations (5), we get:

$$(G^{ab}\_{\mid c} - G^{ad}\mathcal{C}^b\_{dc} - G^{bd}\mathcal{C}^a\_{dc})\xi^a\_a\xi^\beta\_b = 0.$$

The Hamilton–Jacobi equation for a charged test-particle in an external electromagnetic field with potential *Ai* is:

$$H = \mathbf{g}^{i\bar{j}} P\_{\bar{i}} P\_{\bar{j}} = m, \quad P\_{\bar{i}} = p\_{\bar{i}} + A\_{i\prime}, \quad p\_{\bar{i}} = \partial\_{\bar{i}} \varrho. \tag{7}$$

The integrals of motion of the free Hamilton–Jacobi equation are given using Killing vector fields as follows:

$$X\_a = \xi\_a^i p\_{i\prime} \tag{8}$$

Thus, the symmetry of the space given by the Killing vector fields is directly related to the symmetry of the equations of the geodesics given by the integrals of motion. The Hamilton–Jacobi method makes it possible to find these integrals and use them to integrate the geodesic equations. Therefore, the study of the behavior of geodesics is necessary for the study of the geometry of space.

The linear momentum integral of Equation (7) has the following form:

$$X\_a = \xi\_a^t P\_i + \gamma\_{a\_{\prime}} \tag{9}$$

where *γα* are some functions of *<sup>u</sup>i*. Equation (7) admits a motion integral of the form (8) if *H* and *X* ˆ *a* commute under Poisson brackets:

$$[H, \hat{X}\_a]\_P = \frac{\partial H}{\partial p\_i} \frac{\partial \hat{X}\_a}{\partial x^i} - \frac{\partial H}{\partial x^i} \frac{\partial \hat{X}\_a}{\partial p\_i} = 0 \rightarrow g^{i\sigma} (\zeta^j\_a F\_{ji} + \gamma\_{a,i}) P\_\sigma = 0. \tag{10}$$

Hence:

$$
\gamma\_{a,i} = \xi\_a^j F\_{i\bar{j}}, \quad F\_{\bar{j}i} = A\_{i,\bar{j}} - A\_{\bar{j},i}.\tag{11}
$$

Thus, the admissible electromagnetic field is determined from Equation (11) (see [41]). In [39,40] it was proved that in the case of a homogeneous space, the conditions of (11) can be represented as follows:

$$\mathbf{A}\_{a|b} = \mathbf{C}\_{b\mathbf{u}}^{\varepsilon} \mathbf{A}\_{\varepsilon \prime} \tag{12}$$

at the same time:

$$
\gamma\_a = -\mathbf{A}\_d \to \hat{X}\_d = \xi\_a^\alpha \partial\_\alpha.
$$

Here, it is denoted that:

$$\mathbf{A}\_{\mathfrak{a}} = \xi\_{\mathfrak{a}}^{\vec{\mathfrak{a}}} A\_{\mathfrak{i}\mathfrak{r}}$$

It can be shown that Equation (12) forms a completely integrable system. This system specifies the necessary and sufficient conditions for the existence of algebra of integrals of motion that are linear in momenta for Equation (7). Note that in admissible electromagnetic fields given by the conditions (12), the Klein–Gordon–Fock equation:

$$
\hat{H}\boldsymbol{\varrho} = (\mathbf{g}^{i\hat{\jmath}}\hat{P}\_{\hat{\jmath}}\hat{P}\_{\hat{\jmath}})\boldsymbol{\varrho} = m^2 \boldsymbol{\varrho}, \quad \hat{P}\_{\hat{k}} = \hat{p}\_k + A\_{k\prime} \quad \hat{p}\_k = -i\hat{\nabla}\_k.
$$

also admits an algebra of symmetry operators of the form (see [39,41]):

$$
\pounds\_a = \pounds\_a^i \vplus\_i
$$

∇ ˆ *i* is the covariant derivative operator corresponding to the partial derivative operator— ˆ *∂i* = *ıp*<sup>ˆ</sup>*i* in the coordinate field *<sup>u</sup>i*. Function *ϕ* is a scalar field, *m* = *const*. All admissible electromagnetic fields for the homogeneous spacetime are found in [39]. We will use the results of A.Z. Petrov [37]. We follow the notation used in this book with minor exceptions. For example, the nonignorable variable *x*4 will be denoted *u*0, etc.

#### **3. Maxwell's Equations for an Admissible Electromagnetic Field in Homogeneous Spacetime**

Consider Maxwell's equations with zero electromagnetic field sources in homogeneous spacetime in the presence of an admissible electromagnetic field:

$$\frac{1}{\sqrt{-\mathcal{S}}}(\sqrt{-\mathcal{g}}F^{ij})\_{,j} = 0, \quad \mathcal{g} = \det|\mathcal{g}\_{\alpha\beta}|.\tag{13}$$

when *i* = 0 from the system (13), the equation follows:

$$\frac{1}{\sqrt{-g}}(\sqrt{-g}g^{\alpha\beta}A\_{\beta,0})\_{,a} = 0.\tag{14}$$

Using the Killing Equations (4) and (5), we can obtain:

$$\frac{\mathcal{S}\_{|a}}{\mathcal{S}} = 2\zeta\_{a,a}^{ra}$$

Indeed,

$$-\frac{\mathcal{S}\_{|a}}{\mathcal{S}} = \mathcal{g}\_{|a}^{a\mathcal{B}} \mathcal{g}\_{a\mathcal{B}} = \mathcal{G}\_{|a}^{bc} \mathcal{G}\_{bc} + 2\mathcal{g}\_{a,a}^{a} + 2\mathcal{C}\_{a} = 2\mathcal{g}\_{a,a}^{a} \quad (\mathcal{C}\_{a} = \mathcal{C}\_{ab}^{b}).$$

Substituting this expression and the relation (12) into Equation (14), we get:

$$\mathbf{G}^{ab}\mathbf{C}\_b\mathbf{A}\_{a,0} = 0.\tag{15}$$

In the case of spaces with groups *<sup>G</sup>*3(*I*), *<sup>G</sup>*3(*<sup>I</sup> I*), *<sup>G</sup>*3(*VIII*), *<sup>G</sup>*3(*IX*)*Ca* = 0. That is why Equation (15) is satisfied. In the case of the groups *<sup>G</sup>*3(*III*), −*G*3(*VII*) *Ca* = *constδa*3, and from (15) it follows:

$$
\eta^{3a} \mathbf{A}\_{a,0} = 0, \quad \mathbf{A}\_a = A\_a e^a\_a. \tag{16}
$$

For *i* = *α* we have:

$$\frac{1}{\sqrt{\mathcal{S}}}(\sqrt{\mathfrak{g}}\,\mathrm{g}^{a\beta}F\_{\beta\mathbf{0}})\,\mathrm{.}\rho+\frac{1}{\sqrt{\mathcal{S}}}(\sqrt{\mathfrak{g}}\,\mathrm{g}^{a\beta}\,\mathrm{g}^{\gamma\sigma}F\_{\beta\sigma})\,\mathrm{.}\gamma=0.\tag{17}$$

We transform Equation (17) using the (2) frame. The first term then has the form:

$$-\frac{1}{\sqrt{-g}}(\sqrt{-g}g^{a\beta}F\_{\beta 0})\_{,0} = -\frac{1}{l\_0}(l\_0\eta^{ab}\mathbf{\bar{A}}\_{a,0})\_{,0}\mathfrak{e}^a\_{b,} \quad (l\_0)^2 = \det|\eta\_{ab}|.$$

The second term using the (3) frame, the relations (12), and the commutation relations between the operators of the group can be reduced to the following form:

$$\frac{1}{\sqrt{\mathcal{S}}} (\sqrt{\mathcal{S}} \mathcal{g}^{a\beta} \mathcal{g}^{\gamma \sigma} F\_{\beta \sigma})\_{;\gamma} = \frac{1}{2} G^{a\_2 b\_1} \mathcal{C}^{a}\_{a\_2 b\_2} (2 \mathcal{C}\_{b\_1} G^{b b\_2} + \mathcal{C}^{b}\_{a\_1 b\_1} G^{a\_1 b\_2}) \zeta^a\_b \zeta^b\_{\sigma} \mathcal{c}^{\zeta}\_{\beta} \tilde{\mathcal{A}}\_{\epsilon}.$$

So Equation (17) can be written as follows:

$$\frac{1}{r\_0}(l\_0\eta^{ab}\mathbf{A}\_{b,0})\_{,0} = \mathcal{W}^{ba}\mathbf{A}\_{b,} \tag{18}$$

where:

$$\mathcal{W}^{ab} = (\varepsilon^a\_{\beta}\varepsilon^{a\beta}\_{a\_1})(\varepsilon^a\_a\varepsilon^a\_{b\_1})\mathcal{W}^{a\_1b\_1}, \quad \mathcal{W}^{ab} = \frac{1}{2}G^{a\_2b\_1}\mathcal{C}^{a}\_{a\_2b\_2}(2\mathcal{C}\_{b\_1}\mathcal{G}^{bb\_2} + \mathcal{C}^{b}\_{a\_1b\_1}G^{a\_1b\_2}).\tag{19}$$

Then, Maxwell's equations can be represented as follows:

$$
\mathcal{S}^a\_{\mathcal{O}} = l\_0 \tilde{\mathcal{W}}^{ba} \tilde{\mathbf{A}}\_{b\prime} \tag{20}
$$

$$\mathbf{\dot{A}}\_{a,0} = \frac{1}{l\_0} \beta^b \eta\_{ab}. \tag{21}$$

#### **4. Maxwell's Equations for Spaces Type I–VI According to Bianchi Classification**

The group operators in the canonical coordinate set of homogeneous spaces type I–VI according to the Bianchi classification can be represented as follows (see [37]):

$$X\_1 = p\_1, \quad X\_2 = p\_2, \quad X\_3 = (ru^1 + \varepsilon u^2)p\_1 + nu^2p\_2 - p\_3. \tag{22}$$

The values *k ε*, *n* for each group take the following values.) *G*(*I*) : *k* = 0, *ε* = 0, *n* = 0.


*G*(*V I*) :*k* = 1, *ε* = 0, *n* = 2.

> Structural constants can be represented as follows:

$$\mathbb{C}\_{ab}^{c} = k\delta\_1^c(\delta\_a^1 \delta\_b^3 - \delta\_a^3 \delta\_b^1) + (\varepsilon \delta\_1^c + n \delta\_2^c)(\delta\_a^2 \delta\_b^3 - \delta\_a^3 \delta\_b^2) \rightarrow \mathbb{C}\_a = -(k+n)\delta\_a^3 \tag{23}$$

Find the frame vectors [*ξαa* ], [*eαa* ] and their dual vectors [*ξ<sup>a</sup>α*], [*eaα*].

$$
\zeta^a\_a \zeta^b\_a = e^a\_a e^b\_a = \delta^b\_{a'} \quad \zeta^a\_a \zeta^a\_{\beta} = e^a\_a e^a\_{\beta} = \delta^a\_{\beta}.
$$

For this, we use the metrics of homogeneous spaces and the group operators given in [37].

$$
\begin{aligned}
\mathfrak{J}\_{a}^{a} &= \delta\_{a}^{1}\delta\_{1}^{a} + \delta\_{a}^{2}\delta\_{2}^{a} + \delta\_{a}^{3}(\delta\_{1}^{a}(ku^{1} + \epsilon u^{2}) + \delta\_{2}^{a}nu^{2} - \delta\_{3}^{a}), \\
\\
\mathfrak{J}\_{a}^{a} &= \delta\_{1}^{a}\delta\_{a}^{1} + \delta\_{2}^{a}\delta\_{a}^{2} + \delta\_{3}^{a}(\delta\_{a}^{1}(ku^{1} + \epsilon u^{2}) + \delta\_{a}^{2}nu^{2} - \delta\_{a}^{3}), \\
\\
\mathfrak{J}\_{a}^{a} &= \delta\_{a}^{1}\delta\_{1}^{a}\exp(-ku^{3}) + \delta\_{a}^{2}(-\delta\_{1}^{a}\varepsilon u^{3}\exp(-ku^{3}) + \delta\_{2}^{a}\exp(-nu^{2})) + \delta\_{3}^{a}\delta\_{a}^{3}, \\
\\
\mathfrak{e}\_{a}^{a} &= \delta\_{1}^{a}\delta\_{a}^{1}\exp(ku^{3}) + \delta\_{a}^{2}(\delta\_{1}^{a}\varepsilon u^{3}\expnu^{a} + \delta\_{2}^{a}\exp n\mathrm{u}^{2})) + \delta\_{a}^{3}\delta\_{a}^{3}.
\end{aligned} \tag{25}
$$

 .

With these expressions, we find the matrix *W* ˜ *ab* (19).

$$
\tilde{W}^{ab} = \frac{1}{l\_0^2} \left[ \delta\_1^a \delta\_1^b (\varepsilon g\_{11} + \varepsilon (n - k) g\_{12} - k n g\_{22}) \exp(-2n u^3) \right. \tag{26}
$$

$$
$$

$$
\left[ \delta\_1^b (\delta\_1^a \varepsilon u^3 + \delta\_2^a) n (g\_{12} + \varepsilon g\_{11}) \right] + \delta\_1^a (\delta\_1^b \varepsilon u^3 + \delta\_2^b) k (g\_{12} - \varepsilon g\_{11}) \left].
$$

Here (see [37]):

*eαa*

$$\mathcal{g}\_{11} = a\_{11} \exp 2ku^3, \quad \mathcal{g}\_{12} = (\varepsilon u^3 a\_{11} + a\_{12}) \exp(u+k)u^3, \quad \mathcal{g}\_{22} = (\varepsilon u^3 a\_{11} + 2\varepsilon a\_{12} + a\_{22}) \exp 2nu^3.$$

Maxwell's Equations (20) and (21) become:

$$
\dot{\mathcal{B}}^b = \frac{1}{l\_0} [\delta\_1^a \delta\_1^b (\varepsilon g\_{11} + \varepsilon (n - k) \varrho\_{12} - k \eta\_{22}) \exp(-2nu^3) \tag{27}
$$

$$
\left\{ - (\delta\_1^a \varepsilon u^3 + \delta\_2^a) (\delta\_1^b \varepsilon u^3 + \delta\_2^b) k \eta \varrho\_{11} \exp(-2ku^3) + 

\right.
\tag{28}
$$

$$
\left[ \delta\_1^b (\delta\_1^a \varepsilon u^3 + \delta\_2^a) n (\varrho\_{12} + \varepsilon \varrho\_{11}) \right] + \delta\_1^a (\delta\_1^b \varepsilon u^3 + \delta\_2^b) k (\varrho\_{12} - \varepsilon \varrho\_{11}) \right] \tilde{\mathbf{A}}\_{b\prime}
$$

$$
\mathcal{B}^a = l \eta^{ab} \tilde{\mathbf{A}}\_{b\prime 0}. \tag{28}
$$

 *b*,0.

=

The dots denote the time derivatives. The components **A** ˜ *a* are defined by the solutions of the (12) **A***b* system of equations using the formulas:

$$\mathbf{A}\_d = \varepsilon^a\_a \xi^b\_{\gamma a} \mathbf{A}\_b \tag{29}$$

Further solutions of the system of Equation (27) for homogeneous spaces with groups of motions *<sup>G</sup>*3(*<sup>I</sup>* − *V I*) are given. Spatial metrics are given in the book [37]. Solutions for the system (12) can be found in [38],

$$
\mathfrak{a}\_a = \mathfrak{a}\_a(\mathfrak{u}^0).
$$

*4.1. Group <sup>G</sup>*3(*I*)

As the parameters *k*, *n*,*ε* and *Cabc* equal zero, *<sup>G</sup>*3(*I*) is an Abelian group. The components of the vector electromagnetic potential have the form:

$$\mathbf{A}\_{\mathfrak{a}} = \vec{\mathbf{A}}\_{\mathfrak{a}} = A\_{\mathfrak{a}} = \mathfrak{a}\_{\mathfrak{a}\prime}$$

Substituting these expressions into the system of Equations (27) and (28), we obtain the following system of ordinary differential equations:

$$
\dot{\beta}^a = 0 \to \beta^a = c^a = const;
$$

$$l\_0 \dot{\mathfrak{a}}\_a = a\_{ba} \mathfrak{c}^b \quad \rightarrow \mathfrak{a}\_q = \int \frac{a\_{ab} \mathfrak{c}^b}{l\_0} d\mathfrak{u}^0, \quad l\_0^2 = d \mathfrak{c} t \, |a\_{ab}|.$$

All components of *aab*are arbitrary functions of *u*0.

#### *4.2. Group <sup>G</sup>*3(*<sup>I</sup> I*)

For the group *<sup>G</sup>*3(*<sup>I</sup> I*) the parameters *k*, *n*,*ε* have the following values: *k* = *n* = 0,*ε* = 1. The components of the vector electromagnetic potential in the frames [*ξαa* ] and [*eαa* ] have the form:

$$\mathbf{A}\_1 = \boldsymbol{\alpha}\_1, \quad \mathbf{A}\_2 = \boldsymbol{\alpha}\_2 + \boldsymbol{\alpha}\_1 \boldsymbol{\mu}^3, \quad \mathbf{A}\_3 = \boldsymbol{\alpha}\_1 \boldsymbol{\mu}^3 - \boldsymbol{\alpha}\_3; \quad \mathbf{A}\_a = \boldsymbol{\alpha}\_a.$$

Substituting these expressions into the system of Equations (27) and (28), we obtain the following system of ordinary differential equations:

$$l\_0 \dot{\beta}\_4 = \mathfrak{a}\_1 a\_{11} \delta\_{1a} \to l\_0 \dot{\beta}\_1 = \mathfrak{a}\_1 a\_{11}, \quad \beta\_2 = c\_2, \quad \beta\_3 = c\_3 \quad (\beta\_4 = \delta\_{ab} \beta^b); \tag{30}$$

$$l\_0 \dot{a}\_a = a\_{1a} \beta\_1 + a\_{2a} c\_2 + a\_{3a} c\_3, \quad l\_0^2 = \det|a\_{ab}| \quad (c\_a = const.) \,\text{.}\tag{31}$$

Set of equations(30) and (31) contains five equations for 11 functions:

$$l\_{0\prime} \qquad a\_{ab\prime\prime} \qquad a\_{a\prime} \qquad \beta\_1.$$

We should consider separately the variants *α*1 = 0 and *α*1 = 0.

1. *α*1 = 0 → *β*1 = *c*1 = *const*. Then the set of Equations (30) and (31) has a quadrature solution:

$$\alpha\_{\emptyset} = \int \frac{a\_{\emptyset}c\_{b\_1}\delta^{bb\_1}}{l\_0}d\mu^0 \quad (q=2,3).$$

For *a* = 0, Equation (31) implies a linear dependence of the components *<sup>a</sup>*1*q* :

$$c\_1 a\_{11} + c\_2 a\_{12} + c\_3 a\_{13} = 0.$$

All independent components of *aab* are arbitrary functions of *u*0. 2. *α*1 = 0. Consider the following Equations (30) and (31) from the system:

$$l\_0 \mathfrak{a}\_1 = (a\_{11}\mathfrak{z}\_1 + c\_2 a\_{12} + c\_3 a\_{13}), \quad l\_0 \mathfrak{z}\_1 = a\_{11}\mathfrak{a}\_1. \tag{32}$$

Let us take the function *a*11 out of (32). As a result, we obtain:

$$(\alpha\_1^2 - \beta\_1^{'2})\_{,0} = \frac{2\alpha\_1}{l\_0} (c\_2 a\_{12} + c\_3 a\_{13}).$$

Hence:

$$\beta\_1 = \lg \sqrt{a\_1 \,^2 - 2 \int \frac{a\_1}{l\_0} (c\_2 a\_{12} + c\_3 a\_{13}) d\mu^0} \quad (\zeta^2 = 1).$$

>From the remaining equations of the system, we get:

$$a\_q = \int \frac{(a\_{1q}\beta\_1 + a\_{2q}c\_2 + a\_{3q}c\_3)}{l\_0} du^0 \quad (q = 2, 3); \quad a\_{11} = \frac{l\_0\beta\_1}{a\_1}.$$

The functions *l*0, *α*1, and all components of *aab*, except *a*11, *a*33, are arbitrary functions of *u*0. The component *a*33 results from the equation *l*02 = *det*|*aab*|:

$$a\_{33} = \frac{l\_0^{\cdot 2} + a\_{11}a\_{23}^{\cdot 2} + a\_{22}a\_{13}^{\cdot 2} - 2a\_{12}a\_{13}a\_{23}}{a\_{11}a\_{22} - a\_{12}^{\cdot 2}}\tag{33}$$

.

*4.3. Group <sup>G</sup>*3(*III*)

For the group *<sup>G</sup>*3(*III*) the parameters *k*, *n*,*ε* have the following values: *k* = 1, *n* = *ε* = 0.

The components of the vector electromagnetic potential in the frames [*ξαa* ] and [*eαa* ] have the form:

$$\mathbf{A}\_1 = \mathfrak{a}\_1 \exp \mathfrak{u}^3, \quad \mathbf{A}\_2 = \mathfrak{a}\_2, \quad \mathbf{A}\_3 = \mathfrak{a}\_1 \exp \mathfrak{u}^3 - \mathfrak{a}\_3.$$

Substituting these expressions into the system of Equations (27) and (28), we obtain the following system of ordinary differential equations:

$$l\_0 \dot{\beta}\_4 = \kappa\_1 a\_{12} \delta\_{2a} \to l\_0 \dot{\beta}\_2 = \kappa\_1 a\_{12}, \quad \beta\_1 = c\_1, \quad \beta\_3 = 0; \tag{34}$$

$$l\_0 \mathfrak{a}\_a = a\_{2a} \mathfrak{z}\_2 + a\_{1a} \mathfrak{c}.\tag{35}$$

Here and further, Equation (16) is used, according to which *β*3 = 0. The system of Equations (30) and (31) contains five equations for 11 functions:

$$l\_{0\prime} \quad a\_{ab\prime} \quad a\_{a\prime} \quad \beta\_2.$$

We should separately consider the variants *α*1 = 0 and *α*1 = 0.

1. *α*1 = 0 → *β*2 = *c*2 = *const*. In this case the Then set of equations (30) and (31) has a solution in quadratures:

$$\alpha\_q = \int \frac{a\_{qb} c\_{b\_1} \delta^{bb\_1}}{l\_0} d\mu^0 \quad (q = 2, 3).$$

>From (31) it follows a linear dependence of the components *<sup>a</sup>*1*q* :

$$c\_1 a\_{13} + c\_2 a\_{23} = 0 \to a\_{12} = b a\_{11}, \quad \beta\_1 = b, \quad \beta\_2 = 1.$$

*l*0 and all independent components of *aab* are arbitrary functions of *u*0. The component *a*33 is found from Equation (33).

2. Let *α*1 = 0. Consider the following equations from system (30) and (31):

$$l\_0 \dot{\alpha}\_1 = a\_{12} \beta\_2 + c\_1 a\_{11}, \quad l\_0 \dot{\beta}\_2 = a\_{12} \alpha\_1. \tag{36}$$

.

from system (36), it follows:

$$(\alpha\_1^{\;2} - \beta\_2^{\;2})\_{,0} = \frac{2\alpha\_1}{l\_0}c\_1a\_{11}\dots$$

Hence:

$$
\beta\_2 = \zeta \sqrt{a\_1 ^2 - 2 \int \frac{a\_1}{l\_0} (c\_1 a\_{11} + c\_3 a\_{13}) d\mu^0} (\zeta^2 = 1).
$$

>From the remaining equations of the system, we get:

$$a\_{\eta} = \int \frac{(a\_{2q}\beta\_2 + a\_{1q}c\_1 + a\_{3q}c\_3)}{l\_0} du^0 \quad (q = 2, 3); \quad a\_{11} = \frac{l\_0\dot{\beta}\_2}{a\_1}$$

The functions *l*0, *α*1 and all components of *aab*, except *a*11, *a*33, are arbitrary functions of *u*0. The component *a*33 results from Equation (33).

#### *4.4. Group <sup>G</sup>*3(*IV*)

For the group *<sup>G</sup>*3(*IV*) the parameters *k*, *n*,*ε* have the values: *k* = *n* = *ε* = 1.

The components of the vector electromagnetic potential in the frames [*ξαa* ] and [*eαa* ] have the form:

$$\mathbf{A}\_{1} = \boldsymbol{\alpha}\_{1} \exp \boldsymbol{u}^{3}, \quad \mathbf{A}\_{2} = (\boldsymbol{\alpha}\_{2} + \boldsymbol{\alpha}\_{1} \boldsymbol{u}^{3}) \exp \boldsymbol{u}^{3},$$

$$\mathbf{A}\_{3} = (\boldsymbol{\alpha}\_{1} (\boldsymbol{u}^{1} + \boldsymbol{u}^{2} + \boldsymbol{u}^{2} \boldsymbol{u}^{3}) + \boldsymbol{\alpha}\_{2} \boldsymbol{u}^{2}) \exp \boldsymbol{u}^{3} - \boldsymbol{\alpha}\_{3} \mathbf{y},$$

$$\tilde{\mathbf{A}}\_{4} = \boldsymbol{\alpha}\_{4}.$$

Maxwell's Equations (20) and (21) reduce to the following system:

$$d\_0 \beta\_a = \delta\_{1a} (a\_{11}(a\_1 + a\_2) - a\_1 a\_{22} + a\_2 a\_{12}) + \delta\_{2a} (a\_1 a\_{12} - a\_{11} (a\_1 + a\_2)).\tag{37}$$

$$l\_0 \dot{a}\_a = \beta\_2 a\_{a2} + \beta\_1 a\_{a1}, \quad \beta\_3 = 0. \tag{38}$$

from the system (38) it follows:

$$
\dot{m}\_3 = \int \frac{\beta\_2 a\_{32} + \beta\_1 a\_{31}}{l\_0} du^0. \tag{39}
$$

Let us now consider the remaining equations.

(A) *β*1 = 0.

>From the system (37) it follows:

$$a\_{12} = \frac{1}{\beta\_1} (l\_0 \dot{a}\_2 - \beta\_2 a\_{22}) \quad a\_{11} = \frac{1}{\beta\_1^{-2}} (l\_0 (\dot{a}\_1 \beta\_1 - \dot{a}\_2 \beta\_2) + \beta\_2^{-2} a\_{22}),\tag{40}$$

Using these relations, we obtain a consequence from the remaining equations of the system (37) and (38):

$$
\beta\_1 \beta\_2 - \beta\_2 (\beta\_1 + \beta\_2) = \alpha\_1 \dot{\alpha}\_2 - (\alpha\_1 + \alpha\_2) \dot{\alpha}\_1. \tag{41}
$$

With Equation (41), the dependent functions *αa*, *βa* can be expressed in terms of the independent functions. Let us write down the solutions.

1. 
$$(\alpha\_1 \beta\_1 + \beta\_2 (a\_1 + a\_2)) \beta\_2 \neq 0.$$

$$\beta\_1 = \beta\_2 (b - \ln \beta\_2 - \int \frac{a\_1 \dot{u}\_2 - (a\_1 + a\_2) \dot{u}\_1}{\beta\_2^2} du^0) \beta\_1$$

$$a\_{22} = \frac{l\_0(\dot{\alpha}\_2(\alpha\_1 + \alpha\_2) - \beta\_1(\beta\_1 + \beta\_2))}{\alpha\_1 \beta\_1 + \beta\_2(\alpha\_1 + \alpha\_2)}.$$

*l*0, *a*13, *a*23, *ϕ* are arbitrary functions of time. The function *a*33 is expressed in terms of these functions using the relation (33)

2. *<sup>α</sup>*1*β*1 + *β*2(*<sup>α</sup>*1 + *<sup>α</sup>*2) = 0, *a*22, is an arbitrary function, depending on *u*0.

$$\alpha\_1 = a \exp \, \varphi + b \exp \, \varphi, \quad \alpha\_2 = (1+e)\alpha\_1 \quad \beta\_2 = a \exp \, \varphi - b \exp \, \varphi,$$

$$
\beta\_1 = e \beta\_2 \quad (e = const).
$$

*l*0, *a*13, *a*23, *ϕ* are arbitrary functions of time. The function *a*33 is expressed in terms of these functions using the relation (33).

3. *β*2 = 0.

$$a\_2 = a\_1(a + \ln a\_1), \quad a\_{12} = \frac{l\_0 \dot{a}\_2}{\beta\_1}, \quad a\_{11} = \frac{l\_0 \dot{a}\_1}{\beta\_1}, \quad a\_{22} = \frac{l\_0(\dot{a}\_2(a\_1 + a\_2) - \dot{\beta}\_1 \beta\_1)}{a\_1 \beta\_1}.$$

*l*0, *a*13, *a*23, *α*1, *β*1 are arbitrary functions of time. The function *a*33 is expressed in terms of these functions using the relation (33).

(B) *β*1 = 0. Maxwell's equations take the form:

$$l\_0 \dot{\beta}\_2 = a\_1 a\_{12} - (a\_1 + a\_2) a\_{11}, \quad l\_0 \dot{\beta}\_2 = -a\_1 a\_{22} + (a\_1 + a\_2) a\_{12};$$

$$l\_0 \dot{\alpha}\_1 = \beta\_2 a\_{12}, \quad l\_0 \dot{\alpha}\_2 = \beta\_2 a\_{22}.$$

The set of equations has the following

(a) (*<sup>α</sup>*1 + *<sup>α</sup>*2) = 0.

$$\beta\_2 = \xi \sqrt{b + 2 \int \frac{1}{l\_0} (\dot{a}\_1 (a\_1 + a\_2) - a\_1 \dot{a}\_2) du\_0}, \quad a\_{12} = \frac{l\_0 \dot{a}\_1}{\beta\_2}, \quad a\_{22} = \frac{l\_0 \dot{a}\_2}{\beta\_2}.$$

$$a\_{11} = \frac{l\_0 (a\_1 \dot{a}\_1 - \beta\_2 \dot{\beta}\_2)}{\beta\_2 (a\_1 + a\_2)}$$

*l*0, *a*13, *a*23, *α*1, *α*2 are arbitrary functions of time. The function *a*33 is expressed in terms of these functions using relation (33).

(b) *α*2 = −*α*1 → *α*1 = *a* exp *ϕ* − *b* exp *ϕ β*2 = *a* exp *ϕ* + *b* exp *ϕ*, *a*12 = *l*0*α*˙ 1 *β*2 *a*22 = *l*0*α*˙ 2 *β*2.

 *l*0, *a*11, *a*13, *a*23, *ϕ*, *β*1 are arbitrary functions of time. The function *a*33 is expressed in terms of these functions using the relation (33).

#### *4.5. Group <sup>G</sup>*3(*V*)

For the group *<sup>G</sup>*3(*V*) the parameters *k*, *n*,*ε* have the values: *k* = *n* = 1,*ε* = 0. The components of the vector electromagnetic potential in the frames [*ξαa* ] and [*eαa* ] have the form:

$$\mathbf{A}\_1 = \mathfrak{a}\_1 \exp u^3, \quad \mathbf{A}\_2 = \mathfrak{a}\_2 \exp u^3, \quad \mathbf{A}\_3 = (\mathfrak{a}\_1 u^1 + \mathfrak{a}\_2 u^2) \exp u^3 - \mathfrak{a}\_3 \mathfrak{a}\_2$$

$$
\mathbf{A}\_a = \mathfrak{a}\_a.
$$

Maxwell's Equation (18) reduces to the following system of equations:

$$d\_0 \mathfrak{a}\_a = \beta\_2 a\_{a2} + \beta\_1 a\_{a1}, \quad \beta\_3 = 0. \tag{42}$$

$$d\_0 \beta\_a = \delta\_{1a} (a\_{12} a\_2 - a\_1 a\_{22}) + \delta\_{2a} (a\_{12} a\_1 - a\_{11} a\_2),\tag{43}$$

Hence:

$$
\dot{a}\_3 = \int \frac{\beta\_2 a\_{32} + \beta\_1 a\_{31}}{l\_0} du^0,
$$

$$
l\_1 = (a\_{11}\beta\_1 + a\_{12}\beta\_2), \quad l\_0 \dot{a}\_2 = (a\_{12}\beta\_1 + a\_{22}\beta\_2). \tag{44}
$$

1. *α*1 = 0. From the set of equations (43) it follows:

*l*0*α*˙ 1

$$a\_{12} = \frac{1}{a\_1}(l\_0 \dot{\beta}\_2 + a\_2 a\_{11}), \quad a\_{22} = \frac{1}{a\_1 \dot{\gamma}^2}(l\_0(\dot{\beta}\_2 a\_2 - \dot{\beta}\_1 a\_1) + a\_{11} a\_2 \,^2). \tag{45}$$

Substituting (45) into (44) , we ge<sup>t</sup> the corollary:

$$
\beta\_1 \dot{\beta}\_2 - \beta\_2 \dot{\beta}\_1 = a\_1 \dot{a}\_2 - a\_2 \dot{a}\_1. \tag{46}
$$

$$a\_{11}(\alpha\_1\beta\_1 + \alpha\_2\beta\_2) = l\_0(\dot{\alpha}\_1\alpha\_1 - \dot{\beta}\_2\beta\_2). \tag{47}$$

>From (46), it follows:

$$\alpha\_2 = \alpha\_1(b + \int \frac{\beta\_1 \beta\_2 - \beta\_2 \beta\_1}{\alpha\_1^2} d\mu^0),$$

Let us consider (48).

(a) *<sup>α</sup>*1*β*1 + *<sup>α</sup>*2*β*2 = 0. Then, we have:

$$a\_{11} = \frac{l\_0(\alpha\_1 \dot{\alpha}\_2 - \alpha\_2 \dot{\alpha}\_1)}{\alpha\_1 \beta\_1 + \alpha\_2 \beta\_2};$$

*l*0, *a*13, *a*23, *α*1, *βa* are arbitrary functions of time. The function *a*33 is expressed in terms of these functions using the relation (33).

(b) *<sup>α</sup>*1*β*1 + *<sup>α</sup>*2*β*2 = 0 → *<sup>α</sup>*1*α*˙ 1 − *β*1*β* ˙ 1 = 0, *<sup>α</sup>*1*α*˙ 2 + *β*2*β* ˙ 1 = 0. >From this, it follows:

$$a\_1 = a \exp \,\varphi + b \exp \,\varphi, \quad \beta\_2 = a \exp \,\varphi - b \exp \,\varphi, \quad a\_2 = -l a\_1, \quad \beta\_1 = l \beta\_2.$$

where *a*, *b*, *l* = *const*, *ϕ* = *<sup>ϕ</sup>*(*u*<sup>0</sup>).

*l*0, *a*11, *a*13, *a*23 are arbitrary functions of time. The function *a*33 is expressed in terms of these functions using the relation (33).

2. *α*1 = 0. From the system (43), it follows:

$$a\_{12} = \frac{l\_0 \beta\_1}{a\_2}, \quad a\_{11} = -\frac{l\_0 \beta\_2}{a\_2}, \quad a\_{22} = \frac{l\_0(\mathbb{A}\_2 a\_2 - \beta\_1 \beta\_1)}{a\_2 \beta\_2}, \quad \beta\_1 = a \beta\_2.$$

here *a* = *const*, *l*0, *a*13, *a*23, *α*2, *β*2 are arbitrary functions of time. The function *a*33 is expressed in terms of these functions using the relation (33).

#### *4.6. Group <sup>G</sup>*3(*<sup>V</sup> I*)

For the group *<sup>G</sup>*3(*<sup>V</sup> I*), the parameters *k*, *n*,*ε* have the following values: *k* = 1 *n* = 2,*ε* = 0. The components of the vector electromagnetic potential in the frames [*ξαa* ] and [*eαa* ] have the form:

$$\mathbf{A}\_1 = \mathfrak{a}\_1 \exp \mathfrak{u}^3, \quad \mathbf{A}\_2 = \mathfrak{a}\_2 \exp 2\mathfrak{u}^3, \quad \mathbf{A}\_3 = \mathfrak{a}\_1 \mathfrak{u}^1 \exp \mathfrak{u}^3 + 2\mathfrak{a}\_2 \mathfrak{u}^2 \exp 2\mathfrak{u}^3 - \mathfrak{a}\_3 \mathfrak{u}^3$$

$$\mathbf{A}\_{\mathfrak{a}} = \mathfrak{a}\_{\mathfrak{a}}.$$

˜

Maxwell's Equation (18) has the form:

$$d\_0 \dot{a}\_a = \beta\_2 a\_{a2} + \beta\_1 a\_{a1}.\tag{48}$$

$$
\delta\_0 \dot{\beta}\_4 = \delta\_{1a} (a\_{12} a\_2 - 2a\_1 a\_{22}) + \delta\_{2a} (a\_{12} a\_1 - 2a\_{11} a\_2), \quad \beta \gg 0,\tag{49}
$$

and from the system (48), it follows:

$$
\dot{a}\_3 = \int \frac{\beta\_2 a\_{32} + \beta\_1 a\_{31}}{l\_0} du^0.
$$

$$
l\_0 \dot{a}\_1 = (a\_{11}\beta\_1 + a\_{12}\beta\_2), \quad l\_0 \dot{a}\_2 = (a\_{12}\beta\_1 + a\_{22}\beta\_2). \tag{50}
$$

I *β*1 = 0, from system (48), it follows:

$$a\_{12} = \frac{1}{\beta\_1} (l\_0 \dot{a}\_2 - \beta\_2 a\_{22}), \quad a\_{11} = \frac{1}{\beta\_1^2} (l\_0 (\dot{a}\_1 \beta\_1 - \dot{a}\_2 \beta\_2) + a\_{22} \beta\_2^{-2}).\tag{51}$$

Substituting (51) into (48), we get:

$$a\_{22}(\alpha\_1 \beta\_1 + 2\alpha\_2 \beta\_2) = l\_0(\alpha\_2 \beta\_2 - \beta\_1 \beta\_1),\tag{52}$$

$$(\left(2a\_1\beta\_1 + a\_2\beta\_2\right)\left(2a\_2\alpha\_1 + \dot{\beta}\_2\beta\_1\right) = \left(\dot{\beta}\_1\beta\_2 + 2\dot{\alpha}\_2\alpha\_1\right)\left(a\_1\beta\_1 + 2a\_2\beta\_2\right) = 0\tag{53}$$

Using this relation, we ge<sup>t</sup> the following solutions:

(1) *<sup>α</sup>*1*β*1 + 2*α*2*β*2 = 0. From (52) it follows:

$$a\_{22} = \frac{l\_0(\dot{\alpha}\_2 \alpha\_2 - \dot{\beta}\_1 \beta\_1)}{(\alpha\_1 \beta\_1 + 2\alpha\_2 \beta\_2)}.$$

Denote:

$$\alpha\_{\emptyset} = a\_{\emptyset} \exp \,\varphi. \quad \beta\_{\emptyset} = b\_{\emptyset} \exp \,\varphi \quad (q = 1, 2),$$

˙

where *aq*, *bq*, *ϕ* are functions of *u*0. From Equation (53), we get:

$$\begin{aligned} \dot{\varphi} &= \frac{(b\_1 b\_2 + 2 \dot{a}\_2 a\_1)(a\_1 b\_1 + 2 a\_2 b\_2) - (2 a\_1 b\_1 + a\_2 b\_2)(2 a\_2 \dot{a}\_1 + b\_2 b\_1)}{(2 a\_1 a\_2 + b\_1 b\_2)(a\_1 b\_1 - a\_2 b\_2)}; \\\\ a\_{12} &= \frac{l\_0 (\dot{\varphi} a\_2 + a\_2) - b\_2 a\_{22}}{b\_1}; \quad a\_{11} = \frac{l\_0 ((a\_1 b\_1 - a\_2 b\_2) \dot{\varphi} + \dot{a}\_1 b\_1 - \dot{a}\_2 b\_2) + b\_2^2 a\_{22}}{b\_1^2}; \\\\ a\_{22} &= \frac{l\_0 ((a\_2^2 - b\_1^2) \dot{\varphi} + \dot{a}\_2 a\_2 - \dot{b}\_1 b\_1)}{2 a\_1 b\_1 + a\_2 b\_2}. \end{aligned}$$

 *l*0, *a*13, *a*23, *aq*, *bq* are arbitrary functions dependent on time. The function *a*33 is expressed by these functions using the relation (33)

(2) *α*˙ 2*α*2 − *β* ˙ 1*β*1 = 0 → *<sup>α</sup>*1*β*1 + 2*α*2*β*2 = 0. *<sup>a</sup>*22—is an arbitrary function from *u*0;

$$
\alpha\_2 = a \exp \, \varphi - b \exp(-\varphi), \quad \beta\_1 = a \exp \, \varphi + b \exp(-\varphi).
$$

>From this, it follows:

˙

(a)

$$a\_1 = -\frac{\beta\_2}{2} (\frac{a \exp \, q - b \exp(-q)}{a \exp \, q + b \exp(-q)});$$

$$a\_{12} = l\_0 \dot{q} - \frac{\beta\_2 a\_{22}}{\beta\_1}, \quad a\_{11} = \frac{l\_0 (\dot{a}\_1 \beta\_1 - \dot{a}\_2 \beta\_2) + \beta\_2^2 a\_{22}}{\beta\_1^2}$$

(b) *ϕ*˙ = 0

*β*1 = 1, *α*2 = <sup>−</sup>2*b*, *α*1 = −*bβ*2, *a*12 = −*β*2*a*22, *a*11 = −*bl*0*β* ˙ 2 + *β*22*a*22.

> where *l*0, *<sup>a</sup>*.*b* = *consta*22, *a*13, *a*23, *β*2, *ϕ* are arbitrary functions dependent on time.

II *β*1 = 0.

>From (48) and (49) it follows:

$$a\_{12} = \frac{2l\_0 \dot{a}\_2 a\_2}{\beta\_2}, \quad a\_{22} = \frac{l\_0 \dot{a}\_2}{\beta\_2}, \quad a\_{11} = \frac{l\_0 (2b^2 \dot{a}\_2 a\_2^{-3} - \beta\_2 \dot{\beta}\_2)}{2a\_2 \beta\_2}, \qquad a\_1 = ba\_2^{-2}.\tag{54}$$

*l*0, *a*22, *a*13, *a*23, *<sup>α</sup>*2*β*2 depends arbitrarily on time functions. The function *a*33 is expressed in terms of these functions using the relation (33).
