  We obtain exhaustive classification of inequivalent realizations of the Witt
and Virasoro algebras by Lie vector fields of differential operators in the
space $\mathbb{R}^3$. Using this classification we describe all inequivalent
realizations of the direct sum of the Witt algebras in $\mathbb{R}^3$. These
results enable constructing all possible (1+1)-dimensional classically
integrable equations that admit infinite dimensional symmetry algebra
isomorphic to the Witt or the direct sum of Witt algebras. In this way the new
classically integrable nonlinear PDE in one spatial dimension has been
obtained. In addition, we construct a number of new nonlinear (1+1)-dimensional
PDEs admitting infinite symmetries.
