  The vacuum Robinson-Trautman solution admits a shear-free and twist-free null
geodesic congruence with a nonvanishing expansion. We perform a comprehensive
classification of solutions exhibiting this property in Einstein's gravity with
a massless scalar field, assuming that the solution belongs at least to
Petrov-type II and some of the components of Ricci tensor identically vanish.
We find that these solutions can be grouped into three distinct classes: (I-a)
a natural extension of the Robinson-Trautman family incorporating a scalar hair
satisfying the time derivative of the Ricci flow equation, (I-b) a novel
non-asymptotically flat solution characterized by two functions satisfying
Perelman's pair of the Ricci flow equations, and (II) a dynamical solution
possessing ${\rm SO}(3)$, ${\rm ISO}(2)$ or ${\rm SO}(1,2)$ symmetry. We
provide a complete list of all explicit solutions falling into Petrov type D
for classes (I-a) and (I-b). Moreover, leveraging the massless solution in
class (I-a), we derive the neutral Robinson-Trautman solution to the ${\cal
N}=2$ gauged supergravity with the prepotential $F(X) =-iX^0X^1$. By flipping
the 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 spacetime
singularity. Although the general solution is unavailable for class (II), we
find a new dynamical solution with spherical symmetry from the AdS-Roberts
solution via AdS/Ricci-flat correspondence.
