  We study the local solvability of a class of operators with multiple
characteristics. The class considered here complements and extends the one
studied in [9], in that in this paper we consider some cases of operators with
complex coefficients that were not present in [9]. The class of operators
considered here ideally encompasses classes of degenerate parabolic and
Schr\"odinger type operators. We will give local solvability theorems. In
general, one has $L^2$ local solvability, but also cases of local solvability
with better Sobolev regularity will be presented.
