Some generalizations of Halphen's equations
The solutions of an ordinary differential equation in the complex domain may be multivalued. A semicomplete polynomial vector field on \({\mathbb C}^{n}\) is a polynomial vector field whose every solution is single-valued. The paper under review is devoted to the proof of the following theorem. Theorem. For every \(\rho\in{\mathbb C}\setminus\{0;1\}\), there exists a non empty open bounded subset \(B_{\rho}\subset{\mathbb C}\) such that the vector field \[ \begin{aligned} X_{\rho}(\alpha)&= [ z_{1}^{2}+z_{2}z_{3}(z_{2}-z_{3})(\{\alpha-\rho\}z_{2}+\rho\{1-\alpha\}z_{3})]\frac{\partial}{\partial z_{1}}\\ &+ z_{2}(z_{1}+\rho z_{2}z_{3}-\rho z_{3}^{2})\frac{\partial}{\partial z_{2}}+z_{3}(z_{1}+z_{2}^{2}-z_{2}z_{3})\frac{\partial}{\partial z_{3}} \end{aligned} \] is semicomplete for every \(\alpha\in\overline{B}_{\rho}\). A solution of any such \(X_{\rho}(\alpha)\) is defined in the complement of \(\infty\) in \({\mathbb CP}^{1}\) of an invariant component of the discontinuity domain of either a quasifuschian, a totally degenerate or a cusp Kleinian group (and all these situations appear). The proof is based on Lie theoretic methods connected with the so-called Halphen's equations. Finally an example is provided and some questions are addressed.
scientific article; zbMATH DE number 6006852
scientific article