  We propose a noncommutative version of the Euclidean Lie algebra $E_2$.
Several types of non-Hermitian Hamiltonian systems expressed in terms of
generic combinations of the generators of this algebra are investigated. Using
the breakdown of the explicitly constructed Dyson maps as a criterium, we
identify the domains in the parameter space in which the Hamiltonians have real
energy spectra and determine the exceptional points signifying the crossover
into the different types of spontaneously broken PT-symmetric regions with
pairs of complex conjugate eigenvalues. We find exceptional points which remain
invariant under the deformation as well as exceptional points becoming
dependent on the deformation parameter of the algebra.
