Generalisation of Bureau-Guillot systems with Painlev\'e transcendents in the coefficients

Abstract

We construct a generalisation of what we call Bureau-Guillot systems, i.e. systems of first order equations with coefficient functions being Painlev\'e transcendents. The same Painlev\'e equation is related to the system and it appears as regularising condition in the regularisation process. The systems considered are birationally equivalent to the Okamoto polynomial Hamiltonian systems with rational coefficients for Painlev\'e equations, hence they possess the Painlev\'e property. This work extends the results of Bureau-Guillot in a two-fold way. On one side, we consider polynomial systems with degree larger than 2 that are free of movable critical points. These systems contain not only transcendents $\text{P}_{\text{I}}$ and $\text{P}_{\text{II}}$ in the coefficients, but also transcendents $\text{P}_{\text{III}}$, $\text{P}_{\text{IV}}$, $\text{P}_{\text{V}}$ and $\text{P}_{\text{VI}}$ (and/or their derivatives). On the other side, we explore examples of rational systems with the Painlev\'e transcendents in the coefficients birationally equivalent to the Okamoto polynomial systems. Lastly, we present a simpler version of the change of variables to obtain the analogues of the Bureau-Guillot systems. In this framework we discuss generalisations including the mixed case: systems related to the equation $(\text{P}_{\text{J}})$, with ${\text{J}=\text{I}, \dots, \text{VI}}$, containing coefficient functions that are solutions to $(\text{P}_{\text{K}})$ with $\text{K}\neq \text{J}$. In the latter, the equation $(\text{P}_{\text{K}})$ appears as a regularising condition during the regularisation process. Although we are primarily interested in systems possessing the Painlev\'e property, we also briefly discuss an analogous construction for systems including coefficient functions solving quasi-Painlev\'e equation.