  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 non-trivial 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. Non-integrable modifications of our model
lead to a strong-coupling limit of our metallic phase with a ground state
manifold that shows an extensive entropy.
