Geodesic
On the variational integrating matrix for hyperbolic systems
Fundamentalʹnaâ i prikladnaâ matematika, Tome 12 (2006) no. 7, pp. 251-262.

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We obtain a necessary and sufficient condition for a hyperbolic system to be an Euler–Lagrange system with a first-order Lagrangian up to multiplication by some matrix. If this condition is satisfied and an integral of the system is known to us, then we can construct a family of higher symmetries that depend on an arbitrary function. Also, we consider the systems that satisfy the above criterion and that possess a sequence of the generalized Laplace invariants with respect to one of the characteristics; then we prove that the generalized Laplace invariants with respect to the other characteristic are uniquely defined. 
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S. Ya. Startsev. On the variational integrating matrix for hyperbolic systems. Fundamentalʹnaâ i prikladnaâ matematika, Tome 12 (2006) no. 7, pp. 251-262. http://geodesic.mathdoc.fr/item/FPM_2006_12_7_a16/

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