  There is currently much interest in the two-axis countertwisting spin
squeezing Hamiltonian suggested originally by Kitagawa and Ueda, since it is
useful for interferometry and metrology. No analytical solution valid for
arbitrary spin values seems to be available. In this article we systematically
consider the issue of the analytical solvability of this Hamiltonian for
various specific spin values. We show that the spin squeezing dynamics can be
considered to be analytically solved for angular momentum values upto $21/2$,
i.e. for $21$ spin half particles. We also identify the properties of the
system responsible for yielding analytic solutions for much higher spin values
than based on naive expectations. Our work is relevant for analytic
characterization of squeezing experiments with low spin values, and
semi-analytic modeling of higher values of spins.
