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