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We consider a simple model of a scalar field with $U(1)$ current algebragauge symmetry coupled to $2D$-gravity in order to clarify the origin ofStuckelberg symmetry in the $w_{\infty}$-gravity theory. An analogous symmetrytakes place in our model too. The possible central extension of the completesymmetry algebra and the corresponding critical dimension have been found. Theanalysis of the Hamiltonian and the constraints shows that the generators ofthe current algebra, the reparametrization and the Stuckelberg symmetries arenot independent. The connection of the model with $w_{1+\infty}-$ and$W_{1+\infty}$-gravity is discussed.Comment: INRNE-TH/02/93, LaTeX, 11 pages, (slight change of title and other  minor changes

A U(1) KAC-MOODY MODEL COUPLED TO 2D-GRAVITY