Non-Hermitian but \({\cal PT}-\)symmetric quantum system of an \(N-\)plet of bosons described by the three-parametric Bose-Hubbard Hamiltonian \(H(\gamma,v,c)\) is picked up, in its special exceptional-point limit \(c \to 0\) and \(\gamma \to v\), as an unperturbed part of the family of generalized Bose-Hubbard-like Hamiltonians \(\mathfrak{H}(\lambda)=H(v,v,0)+\lambda\,{\cal V}\) for which the unitarity of the perturbed system is required. This leads to the construction of two different families of Hamiltonians \(\mathfrak{H}(\lambda)\). In the first one the number \(N\) of bosons is assumed conserved while in the second family such an assumption is relaxed. In both cases the anisotropy of the related physical Hilbert space is shown reflected by a highly counterintuitive but operationally realizable structure of admissible perturbations \(\lambda\,{\cal V}\).