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 ≥ 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 Bäcklund
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, Pobořil, 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 zerocurvature
representation of the considered PDEs.