In the framework of the linearized theory of small-amplitude water waves, eigenfunctions of the point spectrum are studied for boundary-value problems in infinite domains. Special types of 3D infinite water pools characterised by cone-shaped bottoms and vertical side walls are considered. By means of the incomplete separation of variables, exploiting the Mellin transform, we reduce construction of the eigenmodes to finding solution for some functional difference equations with meromorphic coefficients. Behaviour of the eigenmodes at the singular point of the boundary and the rate of their decay at infinity are also considered.