Some remarks on high degree polynomial integrals of the magnetic geodesic flow on the two-dimensional torus
This paper aims at fixing a gap in the proof of the main result from [the first two authors, Discrete Contin. Dyn. Syst. 39, No. 11, 6565--6583 (2019; Zbl 1432.37082)]. Consider conformal coordinates \((x,y)\) on the \(2\)-torus with metric \(ds^2=\Lambda(x,y)(dx^2+dy^2)\), and let \[H=\frac{p_x^2+p_y^2}{2 \Lambda(x,y)}.\] The magnetic gedesic flow is the Hamiltonian flow of \(H\) with respect to the magnetic Poisson bracket \[\{-,-\}_{\text{mg}} = \frac{\partial}{\partial x}\wedge \frac{\partial}{\partial p_x}+ \frac{\partial}{\partial y} \wedge \frac{\partial}{\partial p_y} + \Omega(x,y) \frac{\partial}{\partial p_x} \wedge \frac{\partial}{\partial p_y}\,,\] where \(\Omega(x,y)\, dx\wedge dy\) is a closed \(2\)-form defining a fixed nonzero magnetic field. The main result of [loc. cit.] states that if the magnetic geodesic flow has a first integral \(F\) which is polynomial of degree \(N\) in the momenta \((p_x,p_y)\) and which has periodic analytic coefficients at \(N+2\) energy levels \(\{H=E_j\}\), \(E_1,\ldots, E_{N+2}\in \mathbb{R}\) pairwise distinct, then \(\Omega\) and \(\Lambda\) are functions of a unique variable. Furthermore \(H\) admits a first integral \(F_1\) which is linear in momenta. In Section 2 the authors outline the main steps of the proof given in [loc. cit.] and identify an argument which can fail. It is explained in Section 3 how to fix this issue without altering the statement of the important result that was presented.
scientific article; zbMATH DE number 7401155
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