  We study a lattice point counting problem for spheres arising from the
Heisenberg groups. In particular, we prove an upper bound on the number of
points on and near large dilates of the unit spheres generated by the
anisotropic norms $\|(z,t)\|_\alpha = ( |z|^\alpha +
|t|^{\alpha/2})^{1/\alpha}$ for $\alpha\geq 2$. As a first step, we reduce our
counting problem to one of bounding an energy integral. The primary new
challenges that arise are the presence of vanishing curvature and uneven
dilations. In the process, we establish bounds on the Fourier transform of the
surface measures arising from these norms. Further, we utilize the techniques
developed here to estimate the number of lattice points in the intersection of
two such surfaces.
