Accessibility and central integrability in the absence of periodic points

We consider a partially hyperbolic diffeomorphism $f:M\to M$ without periodic points on a closed manifold $M$. We prove that $f$ is accessible when $M$ is a 3-manifold with non-virtually-solvable fundamental group ${\pi }_1(M)$. In the case where $\dim E^c=1$, we demonstrate that the center bundle $E^c$ is uniquely integrable if $f$ lacks accessibility. Furthermore, we provide a complete characterization of accessibility classes for such systems with one-dimensional center bundles.