  We present new family of exact analytic solutions for three anyons in a
harmonic potential (or in free space) in terms of generalized harmonics on
$S^3$, which supplement the known solutions. The new solutions satisfy the
hard-core condition when $\alpha={1\over 3},1$ ($\alpha$ being the statistical
parameter) but otherwise, have finite non-vanishing two-particle colliding
probability density, which is consistent with self-adjointness of the
Hamiltonian. These solutions, however, do not have one-to-one mapping property
between bosonic and fermionic spectra.
