On Radial Distribution and Quasi-exact Solvability of Brioschi-Halphen Equation

The Brioschi-Halphen equation (BHE) is a second order complex differential equation obtained by a two step transformation of the Lamé equation. The Lamé equation is an equation in Astronomical physics used in the study of motion of planetary bodies. In this paper, the radial part of the BHE for sufficiently large $r$ and the argument limit $2\pi$ is obtained. The asymptotic radial wave function associated with BHE is obtained in terms of canonical polynomials ${\mathcal{P}}_{n+1},$ and spherical function in $L^2(G,\mathrm{d}\mu ),G=SL(2,\mathbb{R})$ using point canonical transformation and distributional solution in ${\mathcal{C}}_c^{\mathrm{\infty }}(\mathrm{\Omega })$ using Fourier transform method are obtained.