Partial regularity for quasilinear nonuniformly elliptic systems of the total type

We establish partial $C^{1,\alpha}$-regularity of weak solutions of nonhomogeneous nondiagonal nonuniformly elliptic systems of the type
$$ -\partial/\partial x_\alpha A^i_\alpha(x,u,u_x)=B^i(x,u,u_x),\quad i=1,\dots,N. $$
The typical example of admissible systems is the system of the Euler equations of the variational problem on a minimum of integral $\int_\Omega\mathcal{F}(u_x)d\,x$ with the integrand of the type
$$ \mathcal{F}(p)=a|p|^2+b|p|^m+\sqrt{1+\mathrm{det}^2p},\quad a>0,\ b>0, $$
if $b$ is sufficiently large.