On non-linear partial differential equations with an infinite-dimensional conditional symmetry

The invariance of non-linear partial differential equations under a certain infinite-dimensional Lie algebra AN(z) in N spatial dimensions is studied. The special case A1(2) was introduced in [J. Stat. Phys. 75 (1994) 1023] and contains the Schrödinger Lie algebra sch1 as a Lie subalgebra. It is shown that there is no second-order equation which is invariant under the massless realizations of AN(z). However, a large class of strongly non-linear partial differential equations is found which are conditionally invariant with respect to the massless realization of AN(z) such that the well-known Monge–Ampère equation is the required additional condition. New exact solutions are found for some representatives of this class.