Quantization of two-dimensional affine bodies with stabilized dilatations

Discussed are some quantization problems of two-dimensional affine bodies. Quantum dilatational motion is stabilized by some appropriately chosen model potentials. Isochoric part of the dynamics is geodetic, i.e. potential-free. Surprisingly enough, this is compatible with the existence of discrete spectrum (bounded quantum motion). The Sommerfeld polynomial method is used to perform the quantization of such problems.