  Assisted by general symmetry arguments and a many-body invariant, we
introduce a phase of matter that constitutes a topological SO(5) superfluid.
Key to this finding is the realization of an exactly solvable model that
displays some similarities with a minimal model of superfluid $^3$He. We study
its quantum phase diagram and correlations, and find exotic superfluid as well
as metallic phases in the repulsive sector. At the critical point separating
trivial and nontrivial superfluid phases, our Hamiltonian reduces to the
globally SO(5)-symmetric Gaudin model with a degenerate ground manifold that
includes quartet states. Most importantly, the exact solution permits
uncovering of an interesting non-pair-breaking mechanism for superfluids
subject to external magnetic fields. Nonintegrable modifications of our model
lead to a strong-coupling limit of our metallic phase with a ground-state
manifold that shows an extensive entropy.
