  We prove that the scalar and $2\times 2$ matrix differential operators which
preserve the simplest scalar and vector-valued polynomial modules in two
variables have a fundamental Lie algebraic structure. Our approach is based on
a general graphical method which does not require the modules to be irreducible
under the action of the corresponding Lie (super)algebra. This method can be
generalized to modules of polynomials in an arbitrary number of variables. We
give generic examples of partially solvable differential operators which are
not Lie algebraic. We show that certain vector-valued modules give rise to new
realizations of finite-dimensional Lie superalgebras by first-order
differential operators.
