  We consider $2d$ sigma models with a $D=2+N$ - dimensional Minkowski
signature target space metric having a covariantly constant null Killing
vector. These models are UV finite. The $2+N$-dimensional target space metric
can be explicitly determined for a class of supersymmetric sigma models with
$N$-dimensional `transverse' part of the target space being homogeneous
K\"ahler. The corresponding `transverse' sub-theory is an $n=2$ supersymmetric
sigma model with the exact $\gb$-function coinciding with its one-loop
expression. For example, the finite $D=4$ model has $O(3)$ supersymmetric sigma
model as its `transverse' part. Moreover, there exists a non-trivial dilaton
field such that the Weyl invariance conditions are also satisfied, i.e. the
resulting models correspond to string vacua. Generic solutions are represented
in terms of the RG flow in `transverse' theory. We suggest a possible
application of the constructed Weyl invariant sigma models to quantisation of
$2d$ gravity. They may be interpreted as `effective actions' of the quantum
$2d$ dilaton gravity coupled to a (non-conformal) $N$-dimensional `matter'
theory. The conformal factor of the $2d$ metric and $2d$ `dilaton' are
identified with the light cone coordinates of the $2+N$ - dimensional sigma
model.
