We study symmetries of open bosonic systems in the presence of laser pumping. Non-Hermitian Hamiltonians describing these systems can be parity-time ([Formula: see text]) 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 ([Formula: see text]) symmetry. We observe that [Formula: see text]-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 [Formula: see text]-symmetric Hamiltonians. We believe that our results on the [Formula: see text]-symmetric class of bosonic systems and their spectral singularities can lead to new applications inspired by those of the [Formula: see text]-symmetric systems.

Faculty of Physics Adam Mickiewicz University 61 614 Poznań Poland

RCPTM Joint Laboratory of Optics of Palacký University and Institute of Physics of Czech Academy of Sciences 17 listopadu 12 771 46 Olomouc Czech Republic

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