  Over the past two decades, open systems that are described by a non-Hermitian
Hamiltonian have become a subject of intense research. These systems encompass
classical wave systems with balanced gain and loss, semiclassical models with
mode selective losses, and minimal quantum systems, and the meteoric research
on them has mainly focused on the wide range of novel functionalities they
demonstrate. Here, we address the following questions: Does anything remain
constant in the dynamics of such open systems? What are the consequences of
such conserved quantities? Through spectral-decomposition method and explicit,
recursive procedure, we obtain all conserved observables for general
$\mathcal{PT}$-symmetric systems. We then generalize the analysis to
Hamiltonians with other antilinear symmetries, and discuss the consequences of
conservation laws for open systems. We illustrate our findings with several
physically motivated examples.
