  Bethe ansatz solutions of the two-axis countertwisting Hamiltonian for any
(integer and half-integer) $J$ are derived based on the Jordan-Schwinger
(differential) boson realization of the $SU(2)$ algebra after desired Euler
rotations, where $J$ is the total angular momentum quantum number of the
system. It is shown that solutions to the Bethe ansatz equations can be
obtained as zeros of the extended Heine-Stieltjes polynomials. Two sets of
solutions, with solution number being $J+1$ and $J$ respectively when $J$ is an
integer and $J+1/2$ each when $J$ is a half-integer, are obtained. Properties
of the zeros of the related extended Heine-Stieltjes polynomials for
half-integer $J$ cases are discussed. It is clearly shown that double
degenerate level energies for half-integer $J$ are symmetric with respect to
the $E=0$ axis. It is also shown that the excitation energies of the `yrast'
and other `yrare' bands can all be asymptotically given by quadratic functions
of $J$, especially when $J$ is large.
