  Sloshing eigenvalues and eigenfunctions are studied for vertical cylinders of
constant, finite depth occupied by a two-layer fluid. Two families of
eigenfrequencies are obtained in the form expressing them explicitly via the
eigenvalues of the Neumann Laplacian in the two-dimensional domain\, --
\,cylinder's cross-section. Eigenfrequencies belonging to one of the families
behave similar to those that describe sloshing in a homogeneous fluid, whereas
the other family includes a large number of sufficiently small frequencies
provided the ratio of densities is close to unity. Various properties of
eigenfrequencies are investigated for cylinders of arbitrary cross-section;
they include the dependence on the interface depth and the ratio of densities,
the asymptotics of the eigenvalue counting function. The behaviour of
eigenvalues and the corresponding eigenmodes is illustrated by numerical
examples for circular cylinders without and with a radial baffle.
