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The invariance of non-linear partial differential equations under a certain infinite-dimensional Lie algebra UN (z) in N spatial dimensions is studied. The special case U1 (2) was introduced in [J. Stat. Phys. 75 (1994) 1023] and contains the Schroedinger 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 UN (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 UN (z) such that the well-known Monge-AmpeÌEURre equation is the required additional condition. New exact solutions are found for some representatives of this class.