  We consider a semimartingale reflected Brownian motion (SRBM) in a
two-dimensional wedge. Under standard assumptions on the parameters of the
model (opening of the wedge, angles of the reflections on the axes, drift), we
study the algebraic and differential nature of the Laplace transform of its
stationary distribution. We derive necessary and sufficient conditions for this
Laplace transform to be rational, algebraic, differentially finite or more
generally differentially algebraic. These conditions are explicit linear
dependencies among the angles involved in the definition of the model.
  A complicated integral expression is known for this Laplace transform. In the
differentially algebraic case, we go further and provide an explicit,
integral-free expression. In particular, 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
(corresponding to the so-called Dieker-Moriarty conditions on the parameters).
  To prove these results, we start from a functional equation that the Laplace
transform satisfies, to which we apply tools from diverse horizons. To
establish differential algebraicity, a key ingredient is Tutte's invariant
approach, which originates in enumerative combinatorics. It allows us to
express the Laplace transform (or its square) as a rational function of a
certain canonical invariant, a hypergeometric function in our context. To
establish differential transcendence, we turn the functional equation into a
difference equation and apply Galoisian results on the nature of the solutions
to such equations.
