  We investigate the energy spectrum for hybrid mechanical systems described by
non-parity-symmetric quantum Rabi models. A set of analytical solutions in
terms of the confluent Heun functions and their analytical energy spectrum are
obtained. The analytical energy spectrum includes regular and exceptional
parts, which are both confirmed by direct numerical simulation. The regular
part is determined by the zeros of the Wronskian for a pair of analytical
solutions. The exceptional part is relevant to the isolated exact solutions and
its energy eigenvalues are obtained by analyzing the truncation conditions for
the confluent Heun functions. By analyzing the energy eigenvalues for
exceptional points, we obtain the analytical conditions for the
energy-level-crossings, which correspond to two-fold energy degeneracy.
