  In this paper we will first present some results about the local solvability
property of a class of degenerate second order linear partial differential
operators with smooth coefficients. The class under consideration (which in
turn is a generalization of the Kannai operator) exhibits a degeneracy due to
the interplay between the singularity associated with the characteristic set of
a system of vector fields and the vanishing of a function. Afterward we will
also discuss some local solvability results for two classes of degenerate
second order linear partial differential operators with non-smooth coefficients
which are a variation of the main class presented above.
