The task to find multiple solutions of a system of nonlinear equations is tackeled via a homotopy method. Solutions are found by integrating numerically (via a Hermite predictor-corrector) the differential equation arising from differentiating the homotopy equation, parametrized by the arc-length. The part of the method gives a good approximation to each of the zeros of the original equation. This approximation is then used as a starting point in a Newton iteration. The paper concludes with some numerical examples.