An abelian surface with (1, 6)-polarization (Q2864562)

This is a carefully written paper on the algebraic complete integrability of the Dorizzi-Grammaticos-Ramani's system [\textit{B. Dorizzi} et al., ``Painlevé conjecture revisited'', Phys. Rev. Lett. 1539--1541 (1982)]. Recall that a system is algebraic completely integrable in the sense of Adler-van Moerbeke if it can be linearized on a complex algebraic torus \(\mathbb{C}^{n}/\mathrm{lattice}\) (=abelian variety). The invariants (often called first integrals or constants) of the motion are polynomials and the phase space coordinates (or some algebraic functions of these) restricted to a complex invariant variety defined by putting these invariants equals to generic constants, are meromorphic functions on an abelian variety. Moreover, in the coordinates of this abelian variety, the flows (run with complex time) generated by the constants of the motion are straight lines. Using the method of \textit{M. Adler} and \textit{P. van Moerbeke} [Commun. Math. Phys. 113, 659-700 (1987; Zbl 0647.58022)], the authors show that the general fibre of the Hamiltonian system of Dorizzi, Grammaticos and Ramani and its deformation completes to a \((1, 6)\)-polarised abelian surface. More precisely, the system of Dorizzi-Grammaticos-Ramani is algebraic completely integrable. Its general fibre is isomorphic to \(\mathcal{A}\backslash\mathcal{D}\), where \(\mathcal{A}\) is an abelian surface and \(\mathcal{D}\) a curve of geometric genus \(4\) having a singularity of type \(D_4\). The Hamiltonian vector fields extend to linear vector fields on \(\mathcal{A}\) and \(\mathcal{D}\) puts a \((1,6)\)-polarisation on \(\mathcal{A}\). This system has an integrable deformation depending on a parameter \(a\). If \(a=0\), then \(\mathcal{A}\) is isomorphic to the self-product of the elliptic curve with an automorphism of order \(6\).