  The line element of a class of solutions which match to the solution of Kerr
on an oblate spheroid if the two functions $ F(r)$ and $H(r)$ on which it
depends satisfy certain matching conditions is presented. The non vanishing
components of the Ricci tensor $R_{\mu\nu}$, the Ricci scalar $ R$, the second
order curvature invariant $K$, the eigenvalues of the Ricci tensor, the energy
density $\mu$, the tangential pressure $P_{\perp}$, and the quantity
$\mu+P_{\perp}$ are calculated. A function $F(r)$ is given for which $R$ and
$K$ and therefore the solutions are regular. The function $ H(r) $ should be
such that the solution it gives satisfies at least the Weak Energy Conditions
(WEC). Several $H(r)$ are given explicitly for which the resulting solutions
satisfy the WEC and also the Strong Energy Conditions (SEC) and the graphs of
their $\mu$, $P_{\perp}$ and $\mu+P_{\perp}$ for certain values of their
parameters are presented. It is shown that all solutions of the class are
anisotropic fluid solutions and that there are no perfect fluid solutions in
the class.
