  In this paper we show that a system of three fermions is exactly solvable for
the case of a single-$j$ in the presence of an angular momentum-$J$ pairing
interaction. On the basis of the solutions for this system, we obtain new sum
rules for six-$j$ symbols. It is also found that the "non-integer" eigenvalues
of three fermions with angular momentum $I$ around the maximum appear as
"non-integer" eigenvalues of four fermions when $I$ is around (or larger than)
$J_{\rm max}$ and the Hamiltonian contains only an interaction between pairs of
fermions coupled to spin $J=J_{\rm max}=2j-1$. This pattern is also found in
five and six fermion systems. A boson system with spin $l$ exhibits a similar
pattern.
