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. 
@article{FPM_2006_12_7_a16,
     author = {S. Ya. Startsev},
     title = {On the variational integrating matrix for hyperbolic systems},
     journal = {Fundamentalʹna\^a i prikladna\^a matematika},
     pages = {251--262},
     publisher = {mathdoc},
     volume = {12},
     number = {7},
     year = {2006},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FPM_2006_12_7_a16/}
}
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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/