The geometry of Chazy's homogeneous third-order differential equations (Q2895814)

Chazy studied third order differential equations of the form \(\varphi'''=P(\varphi'',\varphi',\varphi),\) where \(P\) is a polynomial of the independent variable. The first step is to consider equations with constant coeffiients with some homogenous condition on \(P\), for instance the following class of equations NEWLINE\[NEWLINE\varphi'''=a_3\varphi^4+a_2\varphi^2\varphi'+a_1(\varphi')^2+\delta\varphi\varphi'',NEWLINE\]NEWLINE where \(a_i\in\mathbb{C}\).NEWLINENEWLINEThe main purpose of the investigation is to find an analog for the third order differential equations of Painlevé's classification of second order differential equations with no movable critical points (that is of the equations that have the Painlevé property). But for the third order equations instead of the equations with the Painlevé property Chazy considered the equations having only single-valued solution.NEWLINENEWLINEChazy found a complete list of such equations [\textit{J. Chazy}, Acta Math. 34, 317--385 (1911; JFM 42.0340.03)].NEWLINENEWLINE In the present paper, the third order equation by setting \((x,y,z)=(\varphi,\varphi',\varphi'')\) is rewritten as the vector field NEWLINE\[NEWLINEV=y\frac{\partial}{\partial x}+z\frac{\partial}{\partial y}+(a_3x^4+a_2x^2y+a_1y^2+\delta x z)\frac{\partial}{\partial z}NEWLINE\]NEWLINE in \(\mathbb{C}^3\). If the original equation has only single-valued solutions then this vector field also has only single valued solutions.NEWLINENEWLINE The orbit is an affine variety \(\Sigma\subset \mathbb{C}^3\). The main result of the paper is to describe the geometry of this variety (the case Chazy \(XII\) for \(k>7\) is not considered). The proof of the main theorem is based on results of \textit{A. Guillot} and \textit{J. Rebelo} [J. Reine Angew. Math. 667, 27--65 (2012; Zbl 1250.32023)].