On the homotopy type of the space of fiberings of $S^1\times S^2$ by simple closed curves

For most aspherical Seifert-fibered 3-manifolds $M$, the space of Seifert fiberings $SF(M)$ is known to have contractible components. It is also known that the space of Hopf fiberings of the three-sphere is noncontractible. We provide the second example of a non-aspherical 3-manifold $M$ such that $SF(M)$ has noncontractible components. In particular, we show that certain components of $SF(S^1\times S^2)$ are homotopy equivalent to a subspace homeomorphic to the identity-based loop space $\mathrm{\Omega }SO(3)$, and we exhibit second homology generators for both connected components of $SF(S^1\times S^2)$.