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The vacuum Robinson-Trautman solution admits a shear-free and twist-free nullgeodesic congruence. We perform a comprehensive classification of solutionsexhibiting this property in Einstein's gravity with a massless scalar field. Wefind that these solutions can be grouped into three distinct classes: (I-a) anatural extension of the Robinson-Trautman family incorporating a scalar hairsatisfying the time derivative of the Ricci flow equation, (I-b) a novelnon-asymptotically flat solution characterized by two functions satisfyingPerelman's pair of the Ricci flow equations, and (II) a dynamical solutionpossessing ${\rm SO}(3)$, ${\rm ISO}(2)$ or ${\rm SO}(1,2)$ symmetry. Weprovide a complete list of all explicit solutions falling into Petrov type Dfor classes (I-a) and (I-b). Moreover, leveraging the massless solution inclass (I-a), we derive the neutral Robinson-Trautman solution to the ${\calN}=2$ gauged supergravity with the prepotential $F(X) =-iX^0X^1$. By flippingthe sign of the kinetic term of the scalar field, the Petrov-D class (I-a)solution leads to a time-dependent wormhole with an instantaneous spacetimesingularity. Although the general solution is unavailable for class (II), wefind a new dynamical solution with spherical symmetry from the AdS-Robertssolution via AdS/Ricci-flat correspondence.

Robinson-Trautman solutions with scalar hair and Ricci flow