Wyman's solution, self-similarity, and critical behavior

The Wyman solution depends on two parameters, the mass M and the scalar charge Σ. If one fixes M to a positive value, say M0, and lets Σ2 take values along the real line, we show that this solution exhibits a type of critical behavior, in analogy with the nonstatic massless scalar field solutions. For Σ2>0 the space-times have naked singularities, for Σ2=0 one has a Schwarzschild black hole of mass M0 and finally for −M02⩽Σ2<0 one has “wormhole-like” solutions. We also show that the Wyman solution is not self-similar, i.e., it does not admit a homothetic Killing vector.