Explicit integrable systems on two dimensional manifolds with a cubic first integral

A few years ago Selivanova gave an existence proof for some integrable models, in fact geodesic flows on two dimensional manifolds, with a cubic first integral. However the explicit form of these models hinged on the solution of a nonlinear third order ordinary differential equation which could not be obtained. We show that an appropriate choice of coordinates allows for integration and gives the explicit local form for the full family of integrable systems. The relevant metrics are described by a finite number of parameters and lead to a large class of models on the manifolds ${\mb S}^2, {\mb H}^2$ and $P^2({\mb R})$ containing as special cases examples due to Goryachev, Chaplygin, Dullin, Matveev and Tsiganov.

💡 Research Summary

The paper addresses a long‑standing gap in the theory of integrable geodesic flows on two‑dimensional manifolds that admit a cubic first integral. Selivanova’s earlier work proved the existence of such systems but left the explicit form of the metric undetermined because it required solving a highly non‑linear third‑order ordinary differential equation (ODE). The authors show that by an appropriate change of local coordinates the ODE simplifies dramatically: it splits into two linear first‑order equations and a single Bernoulli‑type equation, all of which can be integrated in closed form. The integration introduces a finite set of real parameters (typically denoted α, β, γ, δ) that completely characterize the metric.

In the resulting coordinates ((u,v)) the metric takes the rational form

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