Summary: In a class of potentials \(U(x)= A(A\cosh^2 x-\cosh x - 2A) (1+ A\cosh x)^{-2}\) 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 levels. Such a property, exact determination of ground or first excited level in some classes of potentials, is common for one-dimensional models with corresponding symmetric or asymmetric double-well potentials, respectively.