  Regular bouncing solutions in the framework of a scalar-tensor gravity model
were found in a recent work. We reconsider the problem in the Einstein frame
(EF) in the present work. Singularities arising at the limit of physical
viability of the model in the Jordan frame (JF) are either of the Big Bang or
of the Big Crunch type in the EF. As a result we obtain integrable scalar field
cosmological models in general relativity (GR) with inverted double-well
potentials unbounded from below which possess solutions regular in the future,
tending to a de Sitter space, and starting with a Big Bang. The existence of
the two fixed points for the field dynamics at late times found earlier in the
JF becomes transparent in the EF.
