  Stationary and axisymmetric perfect-fluid metrics are studied under the
assumption of the existence of a conformal Killing vector field and in the
general case of differential rotation. The possible Lie algebras for the
conformal group and corresponding canonical line-elements are explicitly given.
It turns out that only four different cases appear, the abelian and other three
called I, II and III. We explicitly find all the solutions in the abelian and I
cases. For the abelian case the general solution depends on an arbitrary
function of a single variable and the perfect fluid satisfies the equation of
state rho = p+const. This class of metrics is the one presented recently by one
of us. The general solution for case I is a new Petrov type D metric, with the
velocity vector outside the 2-space spanned by the two principal null
directions and a barotropic equation of state rho +3p=0. For the cases II and
III, the general solution has been found only under the further assumption of a
natural separation of variables Ansatz. The conformal Killing vectors in the
solutions that come out here are, in fact, homothetic. No barotropic equation
of state exists in these metrics unless for a new Petrov type D solution
belonging to case II and with rho +3p=0 which cannot be interpreted as an
axially symmetric solution and such that the velocity vector points in the
direction of one of the Killing vectors. This solution has the previously
unknown curious property that both commuting Killing vectors are timelike
everywhere.
