We study symmetries of open bosonic systems in the presence of laser pumping. Non-Hermitian Hamiltonians describing these systems can be parity-time ($${{\mathscr{PT}}}$$ PT ) symmetric in special cases only. Systems exhibiting this symmetry are characterised by real-valued energy spectra and can display exceptional points, where a symmetry-breaking transition occurs. We demonstrate that there is a more general type of symmetry, i.e., rotation-time ($${\mathscr{RT}}$$ RT ) symmetry. We observe that $${\mathscr{RT}}$$ RT -symmetric non-Hermitian Hamiltonians exhibit real-valued energy spectra which can be made singular by symmetry breaking. To calculate the spectra of the studied bosonic non-diagonalisable Hamiltonians we apply diagonalisation methods based on bosonic algebra. Finally, we list a versatile set rules allowing to immediately identifying or constructing $${\mathscr{RT}}$$ RT -symmetric Hamiltonians. We believe that our results on the $${\mathscr{RT}}$$ RT -symmetric class of bosonic systems and their spectral singularities can lead to new applications inspired by those of the $${{\mathscr{PT}}}$$ PT -symmetric systems.

We analyse two quantum systems with hidden parity-time ($${\mathscr {P}\mathscr {T}}$$ P T ) symmetry: one is an optical device, whereas another is a superconducting microwave-frequency device. To investigate their symmetry, we introduce a damping frame (DF), in which loss and gain terms for a given Hamiltonian are balanced. We show that the non-Hermitian Hamiltonians of both systems can be tuned to reach an exceptional point (EP), i.e., the point in parameter space at which a transition from broken to unbroken hidden $${\mathscr {P}\mathscr {T}}$$ P T symmetry takes place. We calculate a degeneracy of a Liouvillian superoperator, which is called the Liouvillian exceptional point (LEP), and show that, in the optical domain, LEP is equivalent to EP obtained from the non-Hermitian Hamiltonian (HEP). We also report breaking the equivalence between LEP and HEP by a non-zero number of thermal photons for the microwave-frequency system.

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