We develop a systematic framework for determining the nature of exceptional points of $n\mathrm{th}$ order (${\mathrm{EP}}_n\mathrm{s}$) in non-Hermitian (NH) systems, represented by complex square matrices. By expressing symmetry-preserving perturbations ($\propto \epsilon$) in the Jordan-normal basis of the defective matrix at an ${\mathrm{EP}}_n$, we show that the upper-$\ell$ Hessenberg structure of the perturbation matrix directly dictates the leading-order eigenvalue- and eigenvector-splitting to be $\propto {\epsilon }^{1/\ell }$, when expanded in a Puiseux series. Applying this to three-band NH models invariant under parity (P), charge-conjugation (C), or parity-time-reversal (PT), we find that ${\mathrm{EP}}_3\mathrm{s}$ in P- and C-symmetric systems are restricted to at most $\sim {\epsilon }^{1/2}$ branch points, while PT-symmetric systems generically support ${\mathrm{EP}}_3\mathrm{s}$ with the strongest possible singularities (viz., $\sim {\epsilon }^{1/3}$). We illustrate these results with concrete three-dimensional models in which exceptional curves and surfaces emerge. We further show that fine-tuned perturbations can suppress the leading-order branch point to a less-singular splitting, which have implications for designing direction-dependent EP-based sensors. The Appendices extend the analysis to four-band P- and C-symmetric models, establishing the existence of ${\mathrm{EP}}_4\mathrm{s}$ with $\sim {\epsilon }^{1/4}$ and $\sim {\epsilon }^{1/2}$ singularities, respectively.

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• Exceptional points
• Non-Hermitian systems
• Symmetries