  We prove local solvability for large classes of operators of the form $$
L=\sum_{j,k=1}^{2n}a_{jk}V_jV_k+i\alpha U,$$ where the $V_j$ are left-invariant
vector fields on the Heisenberg group satisfying the commutation relations
$[V_j,V_{j+n}]=U$ for $1\le j\le n$, and where $A=(a_{jk})$ is a complex
symmetric matrix with semi-definite real part. Our results widely extend all of
the results for the case of non-real, semi-definite matrices $A$ known to date,
in particular those obtained recently jointly with F. Ricci under Sj\"ostrand's
cone condition.
