Symmetries of the Darboux equation

This paper establishes the symmetries of Darboux's equations (1882) on tori. We extend Ince's work (1940) by developing new infinite series expansions in terms of Jacobi elliptic functions around each of the four regular singular points of the Darboux equation which are located at the four half-periods of the torus. The symmetry group of the Darboux equation is given by the Coxeter group $B_4\cong G_{\mathrm{I}}\rtimes_{\Gamma} G_{\mathrm{II}}$ where the actions of $G_{\mathrm{I}}$ correspond to the sign changes of the parameters in the solutions of the equations, which have no effective changes on the equation itself, while the actions of $G_{\mathrm{II}}$ permute the four half-periods of the torus, which are described by a short-exact sequence. We are able to clarify the symmetries of the equation when ordered bases of the underlying torus change. Our results show that it is much more transparent to consider the symmetries of the Darboux equation on a torus than on the Riemann sphere $\mathbb{CP}^1$ as was usually considered by earlier researchers such as Maier (2007). We list the symmetry tables of the Darboux equation in both the Jacobian form and Weierstrass form. A consolidated list of 192 solutions of the Darboux equation is also given. We then consider the symmetries of several classical equations (e.g. Lam\'e equation) all of which are special cases of the Darboux equation. Those terminating solutions of the Darboux equation generalize the classical Lam\'e polynomials.