In this paper we propose C-VNS (Continuous variable neighborhood search) method for finding all solutions to a nonlinear system of equations (NSE). We transform the NSE problem into an equivalent optimization problem, and we use a new objective function that allows us to find all zeros. Instead of the usual sum-of-squares objective function, our objective function is presented as the sum of absolute values. Theoretical investigation confirms that our objective function provides more accurate solutions, irrespective of what optimization method is used. We sacrifice the smoothness property to increase precision. Computational analysis on standard test instances shows that our C-VNS based method is more precise and much faster than the two recent methods from the literature we compared with. Moreover, similar conclusions are derived after comparing our C-VNS based heuristic with many other methods from the literature.