  We consider a two - dimensional Minkowski signature sigma model with a $2+N$
- dimensional target space metric having a null Killing vector. It is shown
that the model is finite to all orders of the loop expansion if the dependence
of the ``transverse" part of the metric $\ggij (u,x)$ on the light cone
coordinate $u$ is subject to the standard renormalization group equation of the
$N$ - dimensional sigma model, $ {d\ggij\over du} = \gb_{ij} =R_{ij} + ... $.
In particular, we discuss the `one - coupling' case when $\ggij(u,x)$ is a
metric of an $N$ - dimensional symmetric space $\gij(x)$ multiplied by a
function $f(u)$. The theory is finite if $f(u)$ is equal to the ``running"
coupling of the symmetric space sigma model (with $u$ playing the role of the
RG ``time"). For example, the geometry of space - time with $\gij$ being the
metric of the $N$ - sphere is determined by the form of the $\gb$ - function of
the $O(N+1)$ model. The ``asymptotic freedom" limit of large $u$ corresponds to
the weak coupling limit of small $2+N$ - dimensional curvature. We prove that
there exists a dilaton field which together with the $2+N$ - dimensional metric
solves the sigma model Weyl invariance conditions. The resulting backgrounds
thus represent new tree level string vacua. We also remark on possible
connections with some $2d$ quantum gravity models.
