  We first establish some general results connecting real and complex Lie
algebras of first-order differential operators. These are applied to completely
classify all finite-dimensional real Lie algebras of first-order differential
operators in $R^2$. Furthermore, we find all algebras which are quasi-exactly
solvable, along with the associated finite-dimensional modules of analytic
functions. The resulting real Lie algebras are used to construct new
quasi-exactly solvable Schroedinger operators on $R^2$.
