U.N. Bassey, U.S. Idiong


The Brioschi-Halphen Equation (BHE) is a second order differential equation in the complex domain obtained by a two-step variable transformation of the Lam\'e equation in the Weierstrass form. In this paper, we present a new algebraization of the BHE by writing its operator as an element of the universal enveloping algebra of the Lie algebra $\mathfrak{s}\ell(2,\mathbb{C}).$ We also obtain a Lie algebraic potential from Lie algebraic equation so formed by using a suitable guage transformation. A polynomial solution of the Lie algebraic equation obtained is presented.

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