  Recently a class of inflationary models satisfying the constant rate of roll
constraint, $\ddot{\phi}+(3+\alpha)H\dot{\phi}=0$, has been studied and
compared with the latest cosmological observational data. We consider the
broader class of constraints $
\ddot{\phi}+\frac{\alpha_1}{2}\dot{\phi}+\alpha_2 H\dot{\phi}=0$ and find an
exact particular solution, without initial singularity, which is an attractor
of the dynamics governed by a positive periodic potential $V(\phi)=E+ F \cos
(\Omega \phi )+G \cos (2 \Omega \phi )$. The spectral index $n_s$ and the
tensor to scalar ratio $r$ are consistent with the recent observational data.
