  We establish the existence and find the necessary and sufficient conditions
for a new class of solutions of Locally Rotationally Symmetric spacetimes that
have non vanishing rotation and spatial twist simultaneously. We transparently
show that the existence of such solutions demand non vanishing 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.
