Geodesic
The geometry of Hamiltonian systems of hydrodynamic type. The generalized hodograph method
Izvestiya. Mathematics , Tome 37 (1991) no. 2, pp. 397-419

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It is proved that there exists an infinite involutive family of integrals of hydrodynamic type for diagonal Hamiltonian systems of quasilinear equations; the completeness of the family is also proved, and a basis for it is constructed for Whitham's equation. Higher integrals and symmetries of these systems are found. 
@article{IM2_1991_37_2_a6,
     author = {S. P. Tsarev},
     title = {The geometry of {Hamiltonian} systems of hydrodynamic type. {The} generalized hodograph method},
     journal = {Izvestiya. Mathematics },
     pages = {397--419},
     publisher = {mathdoc},
     volume = {37},
     number = {2},
     year = {1991},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_1991_37_2_a6/}
}
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S. P. Tsarev. The geometry of Hamiltonian systems of hydrodynamic type. The generalized hodograph method. Izvestiya. Mathematics , Tome 37 (1991) no. 2, pp. 397-419. http://geodesic.mathdoc.fr/item/IM2_1991_37_2_a6/