  We consider a class of multiparameter singular Radon integral operators on
the Heisenberg group ${\mathbb H}^1$ where the underlying variety is the graph
of a polynomial. A remarkable difference with the euclidean case, where
Heisenberg convolution is replaced by euclidean convolution, is that the
operators on the Heisenberg group are always $L^2$ bounded. This is not the
case in the euclidean setting where $L^2$ boundedness depends on the polynomial
defining the underlying surface. Here we uncover some new, interesting
phenomena. For example, although the Heisenberg group operators are always
$L^2$ bounded, the bounds are {\it not} uniform in the coefficients of
polynomials with fixed degree. When we ask for which polynoimals uniform $L^2$
bounds hold, we arrive at the {\it same} class where uniform bounds hold in the
euclidean case.
