  We present a systematic framework to construct model Hamiltonians that have
unconventional superconducting pairing states as exact energy eigenstates, by
incorporating multi-body interactions (i.e., interactions among more than two
particles). The multi-body interactions are introduced in a form of the local
density-density coupling in such a way that any pair configuration in real
space has a constant interaction energy by cancelling the two-body and
multi-body interactions. Our approach is applicable to both spinless and
spinful models in any spatial dimensions and on any bipartite lattices,
facilitating an exhaustive extension of Yang's $s$-wave $\eta$-pairing state to
various other unconventional pairing symmetries ($p$-wave, $d$-wave, $f$-wave,
etc.). Particularly, the constructed eigenstates have off-site pairs with
finite center-of-mass momentum, which leads to superconducting states with
either even-parity and spin-triplet or odd-parity and spin-singlet symmetry. We
verify that the two-dimensional spinful Hubbard model on a square lattice with
the multi-body interactions has the spin-triplet $d$-wave pairing state as an
energy eigenstate, which can be regarded as a quantum many-body scar state as
evidenced from the numerical analysis of the pair correlation function,
entanglement entropy, and level statistics. We also discuss other examples,
including spin-triplet $f$-wave pairing states on a honeycomb lattice and
spin-singlet $p$-wave pairing states in a one dimensional chain. These findings
open up the possibility of realizing nonequilibrium unconventional
superconductivity in a long-lived manner protected against thermalization.
