Isolated singularities of solutions to the equation $hbox{div}(a(|Du|)Du)=0$ in the plane

We study the polar singularities of solutions to equations of the form $div(a(|Du|)Du)=0$ in the plane. We assume a degenerate (and singular) ellipticity condition which includes the p-Laplacian as particular case. We represent the complex gradient of the solutions around a non-removable polar singularity $z_0$ in terms of a series expansion of a quasi conformal mapping H. The analysis of the asymptotic behavior of H gives power estimates from above and below for |Du| and we also obtain an upper bound for the index of Du in $z_0$, depending on the rate of growth of |Du|.