  We consider a massless scalar field, conformally coupled to the Ricci scalar
curvature, in the pre-inflation era of a closed FLRW Universe. The scalar field
potential can be of the form of the Coleman-Weinberg one-loop potential, which
is flat at the origin and drives the inflationary evolution. For positive
values of the conformal parameter \xi, less than the critical value xi_c=(1/6),
the model admits exact solutions with non-zero scale factor and zero initial
Hubble parameter. Thus these solutions can be matched smoothly to the so called
Pre-Big-Bang models. At the end of this pre-inflation era one can match
inflationary solutions by specifying the form of the potential and the whole
solution is of the class C^(1).
