Puiseux series about exceptional singularities dictated by symmetry-allowed Hessenberg forms of perturbation matrices

We develop a systematic framework for determining the nature of exceptional points of $n^{\rm th}$ order (EP$_n$s) in non-Hermitian (NH) systems, represented by complex square matrices. By expressing symmetry-preserving perturbations in the Jordan-normal basis of the defective matrix at an EP$_n$, w