We present a global solution for an optimal feedback controller of the underactuated Heisenberg system or nonholonomic integrator. Employing a recently developed technique based on generating functions appearing in the Hamilton-Jacobi theory, we circumvent a singularity caused by underactuation to develop a nonlinear optimal feedback control in an implicitly analytical form. The systematic procedure to deal with underactuation indicates that generating functions should be effective tools for solving general underactuated optimal control problems.