# Seminar talk, 29 March 2017

We consider four three-dimensional equations: (1) the rdDym equation ${\displaystyle u_{ty}=u_{x}u_{xy}-u_{y}u_{xx}}$, (2) the 3D Pavlov equation ${\displaystyle u_{yy}=u_{tx}+u_{y}u_{xx}-u_{x}u_{xy}}$; (3) the universal hierarchy equation ${\displaystyle u_{yy}=u_{t}u_{xy}-u_{y}u_{tx}}$, and (4) the modified Veronese web equation ${\displaystyle u_{ty}=u_{t}u_{xy}-u_{y}u_{tx}}$. For each equation, using the know Lax pairs and expanding the latter in formal series in spectral parameter, we construct two infinite-dimensional differential coverings and give a full description of nonlocal symmetry algebras associated to these coverings. For all the four pairs of coverings, the obtained Lie algebras of symmetries manifest similar (but not the same) structures: the are (semi) direct sums of the Witt algebra, the algebra of vector fields on the line, and loop algebras; all of them contain a component of finite grading. We also discuss actions of recursion operators on shadows of nonlocal symmetries.