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The paper is devoted to hyperbolic (generally speaking, non-Lagrangian andnonlinear) partial differential systems possessing a full set of differentialoperators that map any function of one independent variable into a symmetry ofthe corresponding system. We demonstrate that a system has the above propertyif 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 aconsequence, such systems possess both direct and inverse Noether operators (inthe terminology of a work by B. Fuchssteiner and A.S. Fokas who have used theseterms for operators that map cosymmetries into symmetries and performtransformations in the opposite direction). Systems admitting Noether operatorsare not exhausted by Euler-Lagrange systems and the systems with formalintegrals. In particular, a hyperbolic system admits an inverse Noetheroperator if a differential substitution maps this system into a systempossessing an inverse Noether operator.

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