  Open classical and quantum systems have attracted great interest in the past
two decades. These include systems described by non-Hermitian Hamiltonians with
parity-time $(\mathcal{PT})$ symmetry that are best understood as systems with
balanced, separated gain and loss. Here, we present an alternative way to
characterize and derive conserved quantities, or intertwining operators, in
such open systems. As a consequence, we also obtain non-Hermitian or Hermitian
operators whose expectations values show single exponential time dependence. By
using a simple example of a $\mathcal{PT}$-symmetric dimer that arises in two
distinct physical realizations, we demonstrate our procedure for static
Hamiltonians and generalize it to time-periodic (Floquet) cases where
intertwining operators are stroboscopically conserved. Inspired by the Lindblad
density matrix equation, our approach provides a useful addition to the
well-established methods for characterizing time-invariants in non-Hermitian
systems.
