  In this paper, we study the structure of the singular set for a $C^{1}$
smooth surface in the $3$-dimensional Heisenberg group $\boldsymbol{H}_{1}$. We
discover a Codazzi-like equation for the $p$-area element along the
characteristic curves on the surface. Information obtained from this ordinary
differential equation helps us to analyze the local configuration of the
singular set and the characteristic curves. In particular, we can estimate the
size and obtain the regularity of the singular set. We understand the global
structure of the singular set through a Hopf-type index theorem. We also
justify that Codazzi-like equation by proving a fundamental theorem for local
surfaces in $\boldsymbol{H}_{1}$.

Differential Geometry;Analysis of PDEs