2D reductions of the equation u _yy = u _tx + u _yu _xx − u _xu _xy and their nonlocal symmetries

Abstract

Abstract. We consider the 3D equation u yy = u tx + u y u xx − u x u xy and its 2D reductions: (1) u yy = (u y + y)u xx − u x u xy − 2 (which is equivalent to the Gibbons-Tsarev equation) and (2) u yy = (u y + 2x)u xx + (y − u x )u xy − u x . Using reduction of the known Lax pair for the 3D equation, we describe nonlocal symmetries of (1) and (2) and show that the Lie algebras of these symmetries are isomorphic to the Witt algebra.