Total energy potential as a superpotential in integrable cosmological models

We study the class of integrable self-action potentials satisfying the conditions that (1) they spontaneously break the symmetry, (2) they describe the inflation phase with a graceful exit (i.e., with an exit that does not require fine tuning parameters), and (3) they feature several consecutive inflation phases. The first condition ensures that the inflation mechanism is qualitatively consistent with elementary particle physics, the second is a necessary ingredient of any inflation theory, and the third is necessary for a theory to be consistent with contemporary observational data, which support the accelerated expansion of the Universe.