In a class of potentials U(x) = A(Acosh ²x-cosh x-2A)(1 + Acosh x)^-2 the ground level is E = 0, and the corresponding wavefunction is known exactly. This excitation is a manifestation of broken translation symmetry in one-dimensional, nonlinear models with a symmetric double-Morse potential. Broken translation symmetry in a one-dimensional model with an asymmetric double-Morse potential results in an exotic class of double-well potentials with exactly determined first excited level. Such a property, exact determination of ground or first excited level in some classes of potentials, is common for one-dimensional models with, respectively, corresponding symmetric or asymmetric double-well potentials.