  The Lie-algebraic method approximates differential operators that are formal
polynomials of {1,x,d/dx} by linear operators acting on a finite dimensional
space of polynomials. In this paper we prove that the rank of the n-dimensional
representation of the operator K=a_k d^k/dx^k+a_{k+1} d^{k+1}/dx^{k+1}+...
+a_{k+p} d^{k+p}/dx^{k+p} is n-k and conclude that the Lie-algebraic reductions
of differential equations allow to approximate only some of solutions of the
differential equation K[u]=f. We show how to circumvent this obstacle when
solving boundary value problems by making an appropriate change of variables.
We generalize our results to the case of several dimensions and illustrate them
with numerical tests.
