Kuriksha O. Group analysis and exact solutions of systems of equations of mathematical physics, invariant with respect to low-dimensional Lie algebras

Normal forms of systems of two first-order ordinary differential equations, whose right hand sides do not involve independent variable, are investigated. The group classification of such systems is carried out up to the equivalence group, which consists of linear point transformations of dependent variables and nonlinear point transformations of independent variable. Systems of two second-order ordinary differential equations, which are invariant with respect to the real Lie algebras of dimensions no greater than four, are exhaustively described. Differential invariants are used for description of these systems. Group analysis of Frohlich-Peierls Hamiltonian model in nonequilibrium state in the one dimensional case is carried out. Using group reductions exact solutions for this model are found. Using the three-dimensional subalgebras of the Lie algebra of Poincare group an extended class of exact solutions for the field equations of the axion electrodynamics is obtained. These solutions include arbitrary parameters and arbitrary functions as well. The most general solutions include six arbitrary functions.