Symmetry Reduction of a Generalized Complex Euler Equation for a Vector Field

The procedure of constructing linear ansatzes is algorithmized. Some exact solutions of a generalized complex Euler equation for a vector field, invariant under subalgebras of the Poincaré algebra AP (1, 3) are found. In this article, we consider the equation ∂Σk ∂x0 +Σl ∂Σk ∂xl = 0, Σk = Ek + iHk (k, l = 1, 2, 3). (1) It was proposed by W. Fushchych [1] to describe vector fields. This equation can be considered as a complex generalization of the Euler equation for ideal liquid [2]. Equation (1) is equivalent to the system of real equations for ~ E = (E1, E2, E3) and ~ H = (H1,H2,H3):  ∂Ek ∂x0 + El ∂Ek ∂xl −Hl ∂Hk ∂xl = 0, ∂Hk ∂x0 +Hl ∂Ek ∂xl + El ∂Hk ∂xl = 0. (2) It was established in paper [1] that the maximal invariance algebra of system (2) is a 24dimensional Lie algebra containing the affine algebra AIGL(4,R) with the basis elements Pα = ∂ ∂xα (α = 0, 1, 2, 3), Γa0 = −x0 ∂ ∂xa − ∂ ∂Ea , Γ00 = −x0 ∂ ∂x0 + El ∂ ∂El +Hl ∂ ∂Hl (l = 1, 2, 3), Γaa = −xa ∂ ∂xa − Ea ∂ ∂Ea −Ha ∂ ∂Ha (no sum over a), Γ0a = −xa ∂ ∂x0 + (EaEk −HaHk) ∂ ∂Ek + (EaHk +HaEk) ∂ ∂Hk , Γac = −xc ∂ ∂xa − Ec ∂ ∂Ea −Hc ∂ ∂Ha (a 6= c; a, c = 1, 2, 3). (3) The algebra AIGL(4,R) contains as a subalgebra the Poincaré algebra AP (1, 3) with the basis elements J0a = −Γ0a − Γa0, Jab = Γba − Γab, Pα (a, b = 1, 2, 3; α = 0, 1, 2, 3).