The eigenvalue problem in a cylindrical lens geometry is studied. Using a conformal mapping method, the shape of the boundary and the Hamiltonian for a free particle are reduced to those of a two-dimensional problem with circular symmetry. The wave functions are separated into two independent Hilbert subspaces due to the inherent symmetry of the problem. For small geometry deformations, the solutions are found by a specially designed perturbation approach. Comparisons between exact and perturbative solutions are made for different lens parameters. As the symmetry of the lens is reduced, the characteristics of the spectrum and the corresponding spatial properties of the wave functions are studied. Our results provide a family of billiard geometries in which the electronic level spectrum is well characterized. In analyzing the level spacing distribution of the spectrum, a strong deviation from the Poisson and Wigner limiting distributions is found as the boundary geometry changes. This intermediate distribution is indicative of a mixed phase space, also revealed explicitly in the classical Poincaré maps we present.