Liouvillian solutions for second order linear differential equations with Laurent polynomial coefficient

Abstract This paper is devoted to a complete parametric study of Liouvillian solutions the general trace-free second order differential equation with Laurent polynomial coefficient. family equations, for fixed orders at 0 and $$\infty$$ ∞ polynomial, seen as an affine algebraic variety. We prove that set Picard-Vessiot integrable equations in enumerable union subvarieties. compute explicitly its components. give some applications well known subfamilies, such doubly confluent biconfluent Heun theory algebraically solvable potentials Shrödinger equations. Also, auxiliary tool, we improve previously criterium linear admit solution.