  It has been shown [arXiv:2007.00845] that three families of highly symmetric
states are many-body scars for any spin-1/2 fermionic Hamiltonian of the form
$H_0+OT$, where $T$ is a generator of an appropriate Lie group. One of these
families consists of the well-known $\eta$-pairing states. In addition to
having the usual properties of scars, these families of states are insensitive
to electromagnetic noise and have advantages for storing and processing quantum
information. In this paper we show that a number of well-known coupling terms,
such as the Hubbard and the Heisenberg interactions, and the Hamiltonians
containing them, are of the required form and support these states as scars
without fine-tuning. The explicit $H_0+OT$ decomposition for a number of most
commonly used models, including topological ones, is provided. To facilitate
possible experimental implementations, we discuss the conditions for the
low-energy subspace of these models to be comprised solely of scars. Further,
we write down all the generators $T$ that can be used as building blocks for
designing new models with scars, most interestingly including the spin-orbit
coupled hopping and superconducting pairing terms. We expand this framework to
the non-Hermitian open systems and demonstrate that for them the scar subspace
continues to undergo coherent time evolution and exhibit the "revivals". A full
numerical study of an extended 2D $tJU$ model explicitly illustrates the novel
properties of the invariant scars and supports our findings.
