  We study the uniqueness of generalized $p$-minimal surfaces in the Heisenberg
group. The generalized $p$-area of a graph defined by $u$ reads $\int |\nabla
u+\vec{F}| + Hu$. If $u$ and $v$ are two minimizers for the generalized
$p$-area satisfying the same Dirichlet boundary condition, then we can only get
$N_{\vec{F}}(u)$ $=$ $N_{\vec{F}}(v)$ (on the nonsingular set) where
$N_{\vec{F}}(w)$ $:=$ $\frac{\nabla w+\vec{F}}{|\nabla w+\vec{F}|}.$ To
conclude $u$ $=$ $v$ (or $\nabla u$ $=$ $\nabla v)$, it is not straightforward
as in the Riemannian case, but requires some special argument in general. In
this paper, for a generalized area functional including $p$-area, we prove that
$N_{\vec{F}}(u)$ $=$ $N_{\vec{F}}(v)$ implies $\nabla u$ $=$ $\nabla v$ in
dimension $\geq $ 3 under some rank condition on derivatives of $\vec{F}$ or
the nonintegrability condition of contact form associated to $u$ or $v$. Note
that in dimension 2 ($n=1),$ the above statement is no longer true. Inspired by
an equation for the horizontal normal $N_{\vec{F}}(u),$ we study the
integrability for a unit vector to be the horizontal normal of a graph. We find
a Codazzi-like equation together with this equation to form an integrability
condition.
