  We prove the existence of a homogeneous singular solution of the critical
equation $$-\Delta u = u^{\frac{Q+2}{Q-2}}$$ on the Heisenberg group $H^n$,
where $Q$ is the \textit{homogeneous dimension}. In order to do this, we
introduce a suitable concept of normal curvature for hypersurfaces. Furthermore
we study the bifurcation of non-homogeneous solutions from the homogeneous one.
