Static spherically symmetric scalar fields in general relativity

In 1957, Bergmann and Leipnik attempted to find static spherically symmetric solutions of a special form of the field equations of general relativity. They were not able to find explicit expressions for the gravitational potentials, and they did not realize that such expressions could be found by using a different coordinate system. Although Buchdahl developed an elegant procedure for finding, by inspection, the solutions sought by Bergmann and Leipnik, his procedure is severely limited when applied to the spherically symmetric case. Indeed, his procedure fails to identify one whole class of such static solutions. The object of this paper is to show that, under the assumptions of Bergmann and Leipnik, the integration of the field equations is almost trivial, and to identify the missing class of solutions.