  It is known that the hyperbolic plane admits a countable infinity of
compactifications into a closed disk such that the isometric action of SL(2;R)
acts analytically on the compactified space. We prove that among those
compactifications, only the two most classical ones (namely the closures of
Poincar\'e's disk and Klein's disk) are algebraic, that is to say obtained as a
union of orbits of a projectivized linear representation of SL(2;R). More
generally, we classify all algebraic actions of SL(2;R) on surfaces.
