  The quasi-Gaudin algebra was introduced to construct integrable systems which
are only quasi-exactly solvable. Using a suitable representation of the
quasi-Gaudin algebra, we obtain a class of bosonic models which exhibit this
curious property. These models have the notable feature that they do not
preserve U(1) symmetry, which is typically associated to a non-conservation of
particle number. An exact solution for the eigenvalues within the quasi-exactly
solvable sector is obtained via the algebraic Bethe ansatz formalism.
