Symmetry reductions and exact solutions of Lax integrable $3$-dimensional systems

Abstract

We present a complete description of $2$-dimensional equations that arise as symmetry reductions of fourf $3$-dimensional Lax-integrable equations: (1) the universal hierarchy equation~$u_{yy}=u_zu_{xy}-u_yu_{xz}$; (2) the 3D rdDym equation $u_{ty}=u_xu_{xy}-u_yu_{xx}$; (3) The basic Veronese web equation $u_{ty}=u_tu_{xy}-u_yu_{tx}$; (4) Pavlov's equation $u_{yy}=u_{tx}+u_yu_{xx}-u_xu_{xy}$.