  The paper concerns the weak differentiability of weak solutions to two kinds
of nonuniform nonlinear degenerate elliptic systems under the $p,q$-growth
condition on the Heisenberg Group. We use the iteration to fractional
difference quotients on the Heisenberg Group to get the weak differentiability
of weak solution $u$ in the vertical direction (i.e., ${L^p}$($1<p<4$)
integrability of $Tu$) and then the second order weak differentiability of weak
solution in the horizontal directions (i.e., ${L^2}$ integrability of
$\nabla_H^2u$) and weak differentiability of gradient of weak solution in the
vertical direction (i.e., ${L^2}$ integrability of $T{\nabla_H}u$)).
