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A new class of completely integrable models is constructed. These models aredeformations of the famous integrable and exactly solvable Gaudin models. Incontrast with the latter, they are quasi-exactly solvable, i.e. admit thealgebraic Bethe ansatz solution only for some limited parts of the spectrum. Anunderlying algebra responsible for both the phenomena of complete integrabilityand quasi-exact solvability is constructed. We call it "quasi-Gaudin algebra"and demonstrate that it is a special non-Lie-algebraic deformation of theordinary Gaudin algebra.Comment: 11 page

QUASI-EXACTLY SOLVABLE DEFORMATIONS OF GAUDIN MODELS AND "QUASI-GAUDIN ALGEBRAS"