  An exact rotating anisotropic fluid solution and a family of exact rotating
anisotropic fluid solutions are presented which satisfy all energy conditions
for certain values of their parameters. The components of the Ricci tensor the
eigenvalues of this tensor and the energy-momentum tensor of the solutions are
given explicitly. All have the ring singularity of Kerr's solution and in
addition the solution one more singularity and some solutions of the family
additional singularities.The solution matches to the extremal solution of Kerr
on two surfaces, which are thin shells and for proper values of the parameters
of the solution approximate oblate spheroids. One of these surfaces has
positive surface density. The solutions of the family satisfy the matching
conditions with the solution of Kerr on two pair of surfaces, which are again
thin shells. The surface density of one pair of surfaces is given explicitly.
Also for proper values of the parameters of the solutions the surfaces of the
other pair approximate oblate spheroids.
