  We consider a $C^{1}$ smooth surface with prescribed $p$(or $H$)-mean
curvature in the 3-dimensional Heisenberg group. Assuming only the prescribed
$p$-mean curvature $H\in C^{0},$ we show that any characteristic curve is
$C^{2}$ smooth and its (line) curvature equals $-H$ in the nonsingular
domain$.$ By introducing characteristic coordinates and invoking the jump
formulas along characteristic curves, we can prove that the Legendrian (or
horizontal) normal gains one more derivative. Therefore the seed curves are
$C^{2}$ smooth. We also obtain the uniqueness of characteristic and seed curves
passing through a common point under some mild conditions, respectively. These
results can be applied to more general situations.
