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We present a one-dimensional multi-component model, known to be partially integrable when restricted to the subspaces made of only two components. By constructing fully anti-symmetrized bases, we find integrable excited eigenstates corresponding to the totally anti-symmetric irreducible representation of the permutation operator in the otherwise non-integrable subspaces. We establish rigorously the breakdown of integrability in those subspaces by showing explicitly the violation of the Yang-Baxter's equation. We further solve the constraints from Yang-Baxter's equation to find exceptional momenta that allows Bethe Ansatz solutions of solitonic bound states. These integrable eigenstates have distinct dynamical consequence from the embedded integrable subspaces previously known, as they do not span their separate Krylov subspaces, and a generic initial state can partly overlap with them and therefore have slow thermalization. However, this novel form of weak ergodicity breaking contrasts that of quantum many-body scars in that the integrable eigenstates involved do not have necessarily low entanglement. Our approach provides a complementary route to arrive at quantum many-body scars since, instead of solving towers of single mode excited states based on a solvable ground state in a non-integrable model, we identify the integrable eigenstates that survive in a deformation of the Hamiltonian away from its integrable point.