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The construction of analytic solutions for quasi-exactly solvable systems isan interesting problem. We revisit a class of models for which the oddsolutions were largely missed previously in the literature: the anharmonicoscillator, the singular anharmonic oscillator, the generalized quantumisotonic oscillator, non-polynomially deformed oscillator, and theSchr\"odinger system from the kink stability analysis of $\phi^6$-type fieldtheory. We present a systematic and unified treatment for the odd and evensectors of these models. We find generic closed-form expressions forconstraints to the allowed model parameters for quasi-exact solvability, thecorresponding energies and wavefunctions. We also make progress in the analysisof solutions to the Bethe ansatz equations in the spaces of model parametersand provide insight into the curves/surfaces of the allowed parameters in theparameter spaces. Most previous analyses in this aspect were on a case-by-casebasis and restricted to the first excited states. We present analysis of thesolutions (i.e. roots) of the Bethe ansatz equations for higher excited states(up to levels $n$=30 or 50). The shapes of the root distributions changedrastically across different regions of model parameters, illustratingphenomena analogous to phase transition in context of integrable models.Furthermore, we also obtain the $sl(2)$ algebraization for the class of modelsin their respective even and odd sectors in a unified way.

Bethe ansatz solutions and hidden $sl(2)$ algebraic structure for a class of quasi-exactly solvable systems