  We present a non-relativistic fermionic field theory in 2-dimensions coupled
to external gauge fields. The singlet sector of the $c=1$ matrix model
corresponds to a specific external gauge field. The gauge theory is
one-dimensional (time) and the space coordinate is treated as a group index.
The generators of the gauge algebra are polynomials in the single particle
momentum and position operators and they form the group $W^{(+)}_{1+\infty}$.
There are corresponding Ward identities and residual gauge transformations that
leave the external gauge fields invariant. We discuss the realization of these
residual symmetries in the Minkowski time theory and conclude that the
symmetries generated by the polynomial basis are not realized. We motivate and
present an analytic continuation of the model which realises the group of
residual symmetries. We consider the classical limit of this theory and make
the correspondence with the discrete states of the $c=1$ (Euclidean time)
Liouville theory. We explain the appearance of the $SL(2)$ structure in
$W^{(+)}_{1+\infty}$. We also present all the Euclidean classical solutions and
the classical action in the classical phase space. A possible relation of this
theory to the $N=2$ string theory and also self-dual Einstein gravity in
4-dimensions is pointed out.
