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In Euclidean $3$-space, it is well known that the Sine-Gordon equation wasconsidered in the nineteenth century in the course of investigations ofsurfaces of constant Gaussian curvature $K=-1$. Such a surface can beconstructed from a solution to the Sine-Gordon equation, and vice versa. Withthis as motivation, employing the fundamental theorem of surfaces in theHeisenberg group $H_{1}$, we show in this paper that the existence of aconstant $p$-mean curvature surface (without singular points) is equivalent tothe existence of a solution to a nonlinear second-order ODE (1.2), which is akind of {\bf Li\'{e}nard equations}. Therefore, we turn to investigate thisequation. 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 $p$-meancurvature surfaces into several classes. As a result, after a kind ofnormalization, we obtain a representation of constant $p$-mean curvaturesurfaces and classify further all constant $p$-mean curvature surfaces. InSection 9, we provide an approach to construct $p$-minimal surfaces. It turnsout that, in some sense, generic $p$-minimal surfaces can be constructed viathis approach. Finally, as a derivation, we recover the Bernstein-type theoremwhich was first shown in [3] (or see [7,8]).

A characterization of constant $p$-mean curvature surfaces in the Heisenberg group $H_1$