We consider an obliquely reflected Brownian motion in a two-dimensional cone. The main goal is to study the algebraic nature of the Laplace transform of its stationary distribution in the recurrent case. We derive necessary and sufficient conditions for the Laplace transform to be differentially algebraic, D-finite algebraic or rational. These conditions are algebraic dependencies among the parameters of the model (drift, opening of the wedge, angles of the reflections on the axes). A complicated integral expression is known for the Laplace transform. In the differentially algebraic case, we go further and compute an explicit, integral-free expression. As a consequence we obtain new derivations of the Laplace transform in several well known cases, namely the skew-symmetric case, the orthogonal reflections case and the sum-of-exponential densities case. To prove these results, we apply different tools to a kernel functional equation that the Laplace transform satisfies. A key ingredient is Tutte’s invariant approach, which allows us to express the Laplace transform as a rational function of a certain minimal invariant.