**Appendix A**

In this Appendix, we present the Tables A1–A6, which are referenced in the main article.



**Table A2.** Lie invariants for the optimal system of the nonrotating system.


*Symmetry* **2019**, *11*, 1115


**Table A3.** Lie invariants for the optimal system of the nonrotating system.

*Ad e***(***εYi***)**  **Y**4 *Y*1 *Y*2 cos *ε* − *Y*3 sin *ε Y*2 sin *ε* + *Y*3 cos *ε Y*4 *Y*5 cos *ε* + *Y*6 sin *ε Y*6 cos *ε* − *Y*5 sin *ε Y*7 **Y**5 *Y*1 + *fεY*6 *Y*2 *Y*3 *Y*4 − *εY*6 *Y*5 *Y*6 *Y*7 − *ε*(*γ* − 1) *Y*5 **Y**6 *Y*1 − *fεY*5 *Y*2 *Y*3 *Y*4 + *εY*5 *Y*5 *Y*6 *Y*7 − *ε*(*γ* − 1) *Y*6

**Y**7 *Y*1 *e*(*γ*−1)*ε*

*Y*2 *e*(*γ*−1)*ε*

*Y*3 *Y*4 *e*(*γ*−1)*ε*

*Y*5 *e*(*γ*−1)*ε*

*Y*6 *Y*7


**Table A5.** Lie invariants for the optimal system of the rotating system.


