Geodesic
The Liouville Canonical Form for Compatible Nonlocal Poisson Brackets of Hydrodynamic Type and Integrable Hierarchies
Funkcionalʹnyj analiz i ego priloženiâ, Tome 37 (2003) no. 2, pp. 28-40

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We reduce an arbitrary pair of compatible nonlocal Poisson brackets of hydrodynamic type generated by metrics of constant Riemannian curvature (compatible Mokhov–Ferapontov brackets) to a canonical form, find an integrable system describing all such pairs, and, for an arbitrary solution of this integrable system, i.e., for any pair of compatible Poisson brackets in question, construct (in closed form) integrable bi-Hamiltonian systems of hydrodynamic type possessing this pair of compatible Poisson brackets of hydrodynamic type. The corresponding special canonical forms of metrics of constant Riemannian curvature are considered. A theory of special Liouville coordinates for Poisson brackets is developed. We prove that the classification of these compatible Poisson brackets is equivalent to the classification of special Liouville coordinates for Mokhov–Ferapontov brackets.
Keywords: metric of constant curvature, integrable hierarchy, system of hydrodynamic type, bi-Hamiltonian system, compatible metrics, flat pencil of metrics
Mots-clés : compatible Poisson brackets, Poisson bracket of hydrodynamic type, Liouville bracket, Liouville coordinates. 
@article{FAA_2003_37_2_a3,
     author = {O. I. Mokhov},
     title = {The {Liouville} {Canonical} {Form} for {Compatible} {Nonlocal} {Poisson} {Brackets} of {Hydrodynamic} {Type} and {Integrable} {Hierarchies}},
     journal = {Funkcionalʹnyj analiz i ego prilo\v{z}eni\^a},
     pages = {28--40},
     publisher = {mathdoc},
     volume = {37},
     number = {2},
     year = {2003},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FAA_2003_37_2_a3/}
}
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O. I. Mokhov. The Liouville Canonical Form for Compatible Nonlocal Poisson Brackets of Hydrodynamic Type and Integrable Hierarchies. Funkcionalʹnyj analiz i ego priloženiâ, Tome 37 (2003) no. 2, pp. 28-40. http://geodesic.mathdoc.fr/item/FAA_2003_37_2_a3/