  In Euclidean $3$-space, it is well known that the Sine-Gordon equation was
considered in the nineteenth century in the course of investigations of
surfaces of constant Gaussian curvature $K=-1$. Such a surface can be
constructed from a solution to the Sine-Gordon equation, and vice versa. With
this as motivation, employing the fundamental theorem of surfaces in the
Heisenberg group $H_{1}$, we show in this paper that the existence of a
constant $p$-mean curvature surface (without singular points) is equivalent to
the existence of a solution to a nonlinear second-order ODE (1.2), which is a
kind of {\bf Li\'{e}nard equations}. Therefore, we turn to investigate this
equation. It is a surprise that we give a complete set of solutions to (1.2)
(or (1.5)), and hence use the types of the solution to divide constant $p$-mean
curvature surfaces into several classes. As a result, after a kind of
normalization, we obtain a representation of constant $p$-mean curvature
surfaces and classify further all constant $p$-mean curvature surfaces. In
Section 9, we provide an approach to construct $p$-minimal surfaces. It turns
out that, in some sense, generic $p$-minimal surfaces can be constructed via
this approach. Finally, as a derivation, we recover the Bernstein-type theorem
which was first shown in [3] (or see [7,8]).
