Abstract
The complete classification of inequivalent realizations of the Virasoro algebra by Lie vector fields over the three-dimensional field of real numbers is obtained. Based on this result is our group classification of second-order differential equations admitting infinite-dimensional Virasoro algebras. In particular, we derive all inequivalent second-order partial differential equations, which admit the direct sum of the Witt algebras. The two well known examples of equations belonging to this class are the wave and hyperbolic Liouville equations. We prove that there is one more nonlinear differential equation enjoying the same group properties as the wave equation.
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