New class of locally rotationally symmetric spacetimes with simultaneous rotation and spatial twist

We establish the existence and find the necessary and sufficient conditions for a new class of solutions of locally rotationally symmetric spacetimes that have nonvanishing rotation and spatial twist simultaneously. We transparently show that the existence of such solutions demands nonvanishing and bounded heat flux and these solutions are self-similar. We provide a brief algorithm indicating how to solve the system of field equations with the given Cauchy data on an initial spacelike Cauchy surface. Finally we argue that these solutions can be used as a first approximation from spherical symmetry to study rotating, inhomogeneous, dynamic and radiating astrophysical stars.