On nonlocal symmetries for the Krichever--Novikov equation

We construct new infinite hierarchies of nonlocal symmetries and cosymmetries for the Krichever--Novikov equation using the inverse of the fourth-order recursion operator of the latter.Comment: 11 pages, no figure

A Reciprocal Transformation for the Constant Astigmatism Equation

We introduce a nonlocal transformation to generate exact solutions of the constant astigmatism equation $z_{yy} + (1/z)_{xx} + 2 = 0$. The transformation is related to the special case of the famous B\"acklund transformation of the sine-Gordon equation with the B\"acklund parameter $\lambda = \pm1$. It is also a nonlocal symmetry

On construction of symmetries and recursion operators from zero-curvature representations and the Darboux-Egoroff system

The Darboux-Egoroff system of PDEs with any number $n\ge 3$ of independent variables plays an essential role in the problems of describing $n$-dimensional flat diagonal metrics of Egoroff type and Frobenius manifolds. We construct a recursion operator and its inverse for symmetries of the Darboux-Egoroff system and describe some symmetries generated by these operators. The constructed recursion operators are not pseudodifferential, but are Backlund autotransformations for the linearized system whose solutions correspond to symmetries of the Darboux-Egoroff system. For some other PDEs, recursion operators of similar types were considered previously by Papachristou, Guthrie, Marvan, Poboril, and Sergyeyev. In the structure of the obtained third and fifth order symmetries of the Darboux-Egoroff system, one finds the third and fifth order flows of an $(n-1)$-component vector modified KdV hierarchy. The constructed recursion operators generate also an infinite number of nonlocal symmetries. In particular, we obtain a simple construction of nonlocal symmetries that were studied by Buryak and Shadrin in the context of the infinitesimal version of the Givental-van de Leur twisted loop group action on the space of semisimple Frobenius manifolds. We obtain these results by means of rather general methods, using only the zero-curvature representation of the considered PDEs.Comment: 20 pages; v2: minor change