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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 Eulerrotations, where $J$ is the total angular momentum quantum number of thesystem. It is shown that solutions to the Bethe ansatz equations can beobtained as zeros of the extended Heine-Stieltjes polynomials. Two sets ofsolutions, with solution number being $J+1$ and $J$ respectively when $J$ is aninteger and $J+1/2$ each when $J$ is a half-integer, are obtained. Propertiesof the zeros of the related extended Heine-Stieltjes polynomials forhalf-integer $J$ cases are discussed. It is clearly shown that doubledegenerate level energies for half-integer $J$ are symmetric with respect tothe $E=0$ axis. It is also shown that the excitation energies of the `yrast'and other `yrare' bands can all be asymptotically given by quadratic functionsof $J$, especially when $J$ is large.

Exact solution of the two-axis countertwisting Hamiltonian for the half-integer $J$ case