The solution of systems of non-linear algebraic equations with discontinuities in the solution search domain is considered. It is demonstrated that such problems are often very difficult to solve, even with the state of the art numerical solvers, and even when initial guesses close to the solutions are used. The application of constrained solution methods that do not require evaluation of function and derivative values outside of a predefined (feasible) subspace of the variables for solving such systems is considered. An algorithm is presented for identifying and handling of sub-expressions that introduce discontinuities. These are either removed by algebraic manipulations, or defined as boundaries of a feasible subspace. Using the proposed approach, it is demonstrated that a feasible solution for originally unsolvable problems can be found.