# solving a piecewise degenerate system

We are solving a system of equations $f(x)=0$, including non-linear equations and piecewise linear equations. The space is subdivided to polyhedral pieces (cells), in every of them the equations from the second group are simply linear and there is $C^0$-continuous connection of the equations between the cells. From the structure of the problem we can prove that in most of the cells Jacobian of the system $\det J>0$. There are also some cells where $\det J=0$, due to linear dependence of equations in the second group. Solution of the system does not necessary belong to $\det J=0$ cell, but the path of the solver can intersect such cell, leading to divergency. We use Newton’s method with backtracking line search for solution. Is there any modification of the method to overcome this problem?