  In an overall framework of quantum mechanics of unitary systems a rather
sophisticated new version of perturbation theory is developed. What is assumed
is, firstly, that the perturbed Hamiltonians $H=H_0+\lambda V$ are
non-Hermitian and lie close to their unobservable exceptional-point (EP)
degeneracy limit $H_0$. Secondly, in this EP limit, the geometric multiplicity
$L$ of the degenerate unperturbed eigenvalue $E_0$ is assumed, in contrast to
the majority of existing studies, larger than one. Under these assumptions the
method of construction of the bound states is described. Its specific
subtleties are illustrated via the leading-order recipe. The emergence of a
counterintuitive connection between the value of $L$, the structure of the
matrix elements of perturbations, and the possible loss of the stability and
unitarity of the processes of the unfolding of the EP singularity is given a
detailed explanation.
