  We consider a $2d$ sigma model with a $2+N$ - dimensional Minkowski signature
target space metric having a covariantly constant null Killing vector. We study
solutions of the conformal invariance conditions in $2+N$ dimensions and find
that generic solutions can be represented in terms of the RG flow in $N$ -
dimensional ``transverse space'' theory. The resulting conformal invariant
sigma model is interpreted as a quantum action of the $2d$ scalar (``dilaton")
quantum gravity model coupled to a (non-conformal) `transverse' sigma model.
The conformal factor of the $2d$ metric is identified with a light cone
coordinate of the $2+N$ - dimensional sigma model. We also discuss the case
when the transverse theory is conformal (with or without the antisymmetric
tensor background) and reproduce in a systematic way the solutions with flat
transverse space known before.
