Noncompactness property of fibers and singularities of non-Euclidean Kovalevskaya system on pencil of Lie algebras

It is shown that Liouville foliations of the family on non-Euclidean analogs of Kovalevskaya integrable system on a pencil of Lie algebras have both compact and noncompact fibers. A bifurcation of their compact common level surface into a noncompact one exists and has a noncompact singular fiber. In particular, this is true for the non-Euclidean $e(2, 1)$-analogue of the Kovalevskaya case of rigid body dynamics. For the case of nonzero area integral, we prove an effective criterion of existence of a noncompact component of the common level surface of first integrals and Casimir functions.