  A fluid is said to be \emph{scale-invariant} when its interaction and kinetic
energies have the same scaling in a dilation operation. In association with the
more general conformal invariance, scale invariance provides a dynamical
symmetry which has profound consequences both on the equilibrium properties of
the fluid and its time evolution. Here we investigate experimentally the
far-from-equilibrium dynamics of a cold two-dimensional rubidium Bose gas. We
operate in the regime where the gas is accurately described by a classical
field obeying the Gross--Pitaevskii equation, and thus possesses a dynamical
symmetry described by the Lorentz group SO(2,1). With the further
simplification provided by superfluid hydrodynamics, we show how to relate the
evolutions observed for different initial sizes, atom numbers, trap frequencies
and interaction parameters by a scaling transform. Finally we show that some
specific initial shapes - uniformly-filled triangles or disks - may lead to a
periodic evolution, corresponding to a novel type of breather for the
two-dimensional Gross--Pitaevskii equation.
