Summary: A static fourth-order Euler-Bernoulli beam equation, corresponding to a negative fractional power law for the applied load, has been completely integrated in this paper. For this equation the Lie symmetry and the Noether symmetry algebras are isomorphic to \(\mathfrak{sl}(2,\mathbb{R})\). Due to this algebra is nonsolvable, the symmetry reductions that have been employed so far in the literature fail to obtain the complete solution of the equation. A new strategy to obtain a third-order reduction has been performed, which provides, by direct integration, one of the first integrals of the equation. This first integral leads to a one-parameter family of third-order equations which preserves \(\mathfrak{sl}(2,\mathbb{R})\) as symmetry algebra. From these equations, three remaining functionally independent first integrals have been computed in terms of solutions to a linear second-order equation and, as a consequence, the exact general solution has been obtained. As far as we know, this has not been previously reported in the literature. The general solution can be expressed in parametric form in terms of a fundamental set of solutions to a one-parameter family of Schrödinger-type equations.