  We study symmetries of open bosonic systems in the presence of laser pumping.
Non-Hermitian Hamiltonians describing these systems can be parity-time
(${\cal{PT}}$) 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
(${\cal{RT}}$) symmetry. We observe that ${\cal{RT}}$-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 ${\cal{RT}}$-symmetric Hamiltonians. We believe
that our results on the ${\cal{RT}}$-symmetric class of bosonic systems and
their spectral singularities can lead to new applications inspired by those of
the ${\cal{PT}}$-symmetric systems.
