Nonlocal Symmetries of Integrable Linearly Degenerate Equations: A Comparative Study

Abstract

We continue the study of Lax integrable equations. We consider four three-dimensional equations: (1) the rdDym equation u_ty = u_xu_xy − u_yu_xx, (2) the Pavlov equation u_yy = u_tx + u_yu_xx − u_xu_xy, (3) the universal hierarchy equation u_yy = u_tu_xy − u_yu_tx, and (4) the modified Veronese web equation u_ty = u_tu_xy − u_yu_tx. For each equation, expanding the known Lax pairs in formal series in the spectral parameter, we construct two differential coverings and completely describe the nonlocal symmetry algebras associated with these coverings. For all four pairs of coverings, the obtained Lie algebras of symmetries manifest similar (but not identical) structures; they are (semi)direct sums of the Witt algebra, the algebra of vector fields on the line, and loop algebras, all of which contain a component of finite grading. We also discuss actions of recursion operators on shadows of nonlocal symmetries.