Polynomial Hamiltonian systems with movable algebraic singularities (Q327603)

For the rational second-order equation \(y''=R(z,y,y'),\) if every solution is single-valued around its movable singularities, that is, singularities depending on initial values, then the equation is reduced to one of the Painlevé equations or is solvable by known classical functions. For second order equations of certain types, as shown by \textit{S. Shimomura} [Ann. Mat. Pura Appl. (4) 186, No. 2, 267--280 (2007; Zbl 1150.34031); Tohoku Math. J. (2) 60, No. 4, 581--595 (2008; Zbl 1166.34057)] and \textit{G. Filipuk} and \textit{R. G. Halburd} [J. Math. Phys. 50, No. 2, 023509, 18 p. (2009; Zbl 1202.34152)], a movable singularity of each solution reached by analytic continuation along a rectifiable curve is an algebraic branch point. This paper deals with the Hamiltonian system \[ dy_1/dz= \partial H/\partial y_2, \quad dy_2/dz= -\partial H/\partial y_1, \] where \(H=H(z,y_1,y_2)\) is a polynomial in \((y_1,y_2)\) with coefficients analytic in \(z\) in some domain \(\Omega \subset \mathbb{C}\) satisfying certain conditions. Supposing the existence of a formal Puiseux-series solution at each point in \(\Omega\setminus \Phi\), where \(\Phi\) consists of some fixed singularities, the author proves that, for every solution, a singularity reached by analytic continuation along a rectifiable curve is an algebraic branch point, around which it is represented by the convergent Puiseux series. Examples in Section 8 include Hamiltonian systems equivalent to the equations discussed by Filipuk and Halburd [loc. cit.] and to an equation of second order and second degree in \(y\) given in [\textit{C. M. Cosgrove} and \textit{G. Scoufis}, Stud. Appl. Math. 88, No. 1, 25--87 (1993; Zbl 0774.34005)].