  The aim of this paper is to establish regularity for weak solutions to the
nondiagonal quasilinear degenerate elliptic systems related to H\"{o}rmander's
vector fields, where the coefficients are bounded with vanishing mean
oscillation. We first prove $L^p$($p \ge 2$) estimates for gradients of weak
solutions by using a priori estimates and a known reverse H\"{o}lder
inequality, and consider regularity to the corresponding nondiagonal
homogeneous degenerate elliptic systems. Then we get higher Morrey and
Campanato estimates for gradients of weak solutions to original systems and
H\"{o}lder estimates for weak solutions.
