Similarity reduction of a (2+1) Volterra system

The continuous point symmetries of (2 + 1) discrete Volterra system is deri ved and it forms a centreless infinite-dimensional Kac-Moody Virasoro algeb ra. Using the symmetries, a similarity reduction of the (2 + 1) Volterra sy stem to a partial differential-difference equation with two independent var iables is obtained. Also, the Lie point symmetries of the reduced partial d ifferential-difference equation is derived and its similarity reduction to an ordinary differential-difference equation is obtained. The ordinary diff erential-difference equation reduces to the third Painleve equation in the continuum limit. Further, a Lax pair for the reduced equation is derived in dicating its complete integrability.