Space of initial conditions for a cubic Hamiltonian system

In this paper we perform the analysis that leads to the space of initial conditions for the Hamiltonian system q′ = p² + zq + α, p′ = −q² − zp − β, studied by the author in a previous article. By compactifying the phase space of the system from C² to CP² three base points arise in the standard coordinate charts covering the complex projective space. Each of these is removed by a sequence of three blow-ups, a construction to regularise the system at these points. The resulting space, where the exceptional curves introduced after the first and second blow-up are removed, is the so-called Okamoto’s space of initial conditions for this system which, at every point, defines a regular initial value problem in some coordinate chart of the space. The solutions in these coordinates will be compared to the solutions in the original variables.