### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 13 (2017), 034, 20 pages      arXiv:1511.09418      https://doi.org/10.3842/SIGMA.2017.034

### Formal Integrals and Noether Operators of Nonlinear Hyperbolic Partial Differential Systems Admitting a Rich Set of Symmetries

Sergey Ya. Startsev
Institute of Mathematics, Ufa Scientific Center, Russian Academy of Sciences, 112 Chernyshevsky Str., Ufa, Russia

Received September 16, 2016, in final form May 18, 2017; Published online May 27, 2017

Abstract
The paper is devoted to hyperbolic (generally speaking, non-Lagrangian and nonlinear) partial differential systems possessing a full set of differential operators that map any function of one independent variable into a symmetry of the corresponding system. We demonstrate that a system has the above property if and only if this system admits a full set of formal integrals (i.e., differential operators which map symmetries into integrals of the system). As a consequence, such systems possess both direct and inverse Noether operators (in the terminology of a work by B. Fuchssteiner and A.S. Fokas who have used these terms for operators that map cosymmetries into symmetries and perform transformations in the opposite direction). Systems admitting Noether operators are not exhausted by Euler-Lagrange systems and the systems with formal integrals. In particular, a hyperbolic system admits an inverse Noether operator if a differential substitution maps this system into a system possessing an inverse Noether operator.

Key words: Liouville equation; Toda chain; integral; Darboux integrability; higher symmetry; hyperbolic system of partial differential equations; conservation laws; Noether theorem.

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