  Aiming towards understanding the question of the discrete states in the
light-cone gauge in the theory of two-dimensional strings with a linear
background charge term, we study the path-integral formulation of the theory.
In particular, by gauge fixing Polyakov's path-integral expression for the 2-d
strings, we show that the light-cone gauge-fixed generating functional is the
same as the conformal gauge-fixed one and is critical for the same value of the
background charge (Q=2 $\sqrt 2 $). Since the equivalence is shown at the
generating functional level, one expects that the spectra of the two theories
are the same. The zero modes of the ratio of the determinants are briefly
analyzed and it is shown that only the constant mode survives in this
formulation. This is an indication that the discrete states may lie in these
zero modes. This result is not particular to the light-cone gauge, but it holds
for the conformal gauge as well.
