  Describing and achieving `unconventional' superconductivity remains a
forefront challenge in quantum many-body physics. Here we use a unitary
mapping, combined with the well-established properties of the attractive
Hubbard model to demonstrate rigorously a Hamiltonian with a low temperature
pair-density-wave (PDW) phase. We also show that the same mapping, when applied
to the widely accepted properties of the repulsive Hubbard model, leads to a
Hamiltonian exhibiting triplet $d$-wave PDW superconductivity and an unusual
combination of ferro- and antiferro-magnetic spin correlations. We then
demonstrate the persistence of the $d$-wave PDW in a Hamiltonian derived from
the mapping of the extended $t$-$J$ model in the large-$U$ limit. Furthermore,
through strategic manipulation of the nearest-neighbor hopping signs of
spin-down electrons, we illustrate the attainability of PDW superconductivity
at other momenta. The intertwining of different magnetic and exotic pairing
correlations noted here may have connections to experimental observations in
spin-triplet candidates like UTe$_2$.
