  One of the unique features of non-Hermitian~(NH) systems is the appearance of
non-Hermitian degeneracies known as exceptional points~(EPs). The extensively
studied defective EPs occur when the Hamiltonian becomes non-diagonalizable.
Aside from this degeneracy, we show that NH systems may host two further types
of degeneracies, namely, non-defective EPs and ordinary (Hermitian) nodal
points. The non-defective EPs manifest themselves by i) the diagonalizability
of the NH Hamiltonian at these points, ii) the non-diagonalizability of the
Hamiltonian along certain intersections of these points and iii) instabilities
in the Jordan decomposition when approaching the points from certain
directions. We demonstrate that certain discrete symmetries, namely
parity-time, parity-particle-hole, and pseudo-Hermitian symmetry, guarantee the
occurrence of both defective and non-defective EPs. We extend this list of
symmetries by including the NH time-reversal symmetry in two-band systems.
Two-band and four-band models exemplify our findings. Through an example, we
further reveal that ordinary nodal points may coexist with defective EPs in NH
models when the above symmetries are relaxed.
