  We greatly simplify the light-cone gauge description of a relativistic
membrane moving in Minkowski space by performing a field-dependent change of
variables which allows the explicit solution of all constraints and a
Hamiltonian reduction to a $SO(1,3)$ invariant $2+1$-dimensional theory of
isentropic gas dynamics, where the pressure is inversely proportional to
(minus) the mass-density. Simple expressions for the generators of the
Poincar\'e group are given. We also find a generalized Lax pair which involves
as a novel feature complex conjugation. The extension to the supersymmetric
case, as well as to higher-dimensional minimal surfaces of codimension one is
briefly mentioned.
