Free Circle Actions and Positive Ricci Curvature on Manifolds with the Cohomology Ring of $S^2\times S^5$

We classify which of the 672 oriented diffeomorphism types of closed, simply-connected spin 7-manifolds with the cohomology ring of $S^2\times S^5$ admit a free circle action. In addition, we show that whenever such an action exists, there exist infinitely many pairwise non-equivalent free circle ac