1) The paper describes a method for generating new solutions to the Einstein-Maxwell equations coupled to a scalar field using the Kaluza-Klein formulation. 2) Starting from a solution that has a Killing vector field, one can obtain a one-parameter family of new solutions by decomposing the metric in different ways using different Killing vector fields. 3) As an example, the paper generates a new stationary, spherically symmetric solution from the Schwarzschild solution by using different Killing vector fields in the 5-dimensional Kaluza-Klein metric.