  Local solvability and non-solvability are classified for left-invariant
differential operators on the Heisenberg group H_1 of the form
L=P_n(X,Y)+Q(X,Y) where the P_n are certain homogeneous polynomials of order n
greater than or equal to 2 and Q is of lower order with X= \partial_x,
Y=\partial_y+x\partial_w on R^3. We extend previous studies of operators of the
form P_n(X,Y) via representations involving ordinary differential operators
with a parameter.
