**12. Conclusions**

A wide variety of multidimensional completely integrable evolution flows on smooth functional manifolds have been constructed. Our approach was based on a generalized Lie-algebraic Adler–Kostant–Symes scheme, applied to the modified holomorphic current loop algebra G := diff : (T*n* × C) - diff : (T*n* × C)<sup>∗</sup>, the semi-direct sum of the loop Lie algebra diff : (T*n* × C) := *Vect* : (T*n* × C) of vector fields on the T*n* × C, *n* ∈ Z+, and its adjoint space diff : (T*n* × C)<sup>∗</sup>. Its relation to the classical *R*-structure on the loop Lie algebra *Vect* : (T*n* × C) is also discussed. The structure of the corresponding seed elements is analyzed, its multidimensional generalizations are presented. We also demonstrated that the obtained Hamiltonian flows are equivalent to the compatibility conditions for the suitably related Lax–Sato type linear vector field equations. We also mentioned a very interesting Lagrange– d'Alembert type mechanical interpretation, naturally related to the devised Lax–Sato vector field equations and their compatibility conditions. As interesting examples, we constructed new modified spatially four-dimensional Mikhalev–Pavlov and Alonso–Shabat type completely integrable equations, appearing in the study of some differential geometric structures on Riemannian spaces with symmetries.

**Funding:** This research was funded by the Department of Computer Science and Telecommunication of the Cracov University of Technology for a local research gran<sup>t</sup> F-2/370/2018/DS.

**Acknowledgments:** I would like to convey my warm thanks to Gerald A. Goldin for many discussions of the work and instrumental help in editing a manuscript during the XXVIII International Workshop on "Geometry in Physics", held on 30 June–7 July 2019 in Białowieza, Poland. My special ˙ appreciation belongs to Stefan Duplij for friendly encouragemen<sup>t</sup> to write this article and to Joel

Lebowitz for the invitation to take part in the 121-st Statistical Mechanics Conference, held 12–14 May 2019 at the Rutgers University, New Brunswick, NJ, USA. I cordially appreciate Joel Lebowitz, Denis Blackmore and Nikolai N. Bogolubov for instructive discussions, useful comments and remarks on the work during the Conference. My warm acknowledgements also belong to my close collaborators Alex A. Balinsky, Radoslaw Kycia, Yarema A. Prykarpatsky, Valeriy H. Samoilenko for the support during my work on manuscript.

**Conflicts of Interest:** The author declares no conflict of interest.
