  We consider a simple model of a scalar field with $U(1)$ current algebra
gauge symmetry coupled to $2D$-gravity in order to clarify the origin of
Stuckelberg symmetry in the $w_{\infty}$-gravity theory. An analogous symmetry
takes place in our model too. The possible central extension of the complete
symmetry algebra and the corresponding critical dimension have been found. The
analysis of the Hamiltonian and the constraints shows that the generators of
the current algebra, the reparametrization and the Stuckelberg symmetries are
not independent. The connection of the model with $w_{1+\infty}-$ and
$W_{1+\infty}$-gravity is discussed.
