  Exceptional points of order $n$ (EP$n$s) appear in non-Hermitian systems as
points where the eigenvalues and eigenvectors coalesce. They emerge if $2(n-1)$
real constraints are imposed, such that EP2s generically appear in two
dimensions (2D). Local symmetries have been shown to reduce this number of
constraints. In this work, we provide a complete characterization of the
appearance of symmetry-induced higher-order EPs in 2D parameter space. We find
that besides EP2s only EP3s, EP4s, and EP5s can be stabilized in 2D. Moreover,
these higher-order EPs must always appear in pairs with their dispersion
determined by the symmetries. Upon studying the complex spectral structure
around these EPs, we find that depending on the symmetry, EP3s are accompanied
by EP2 arcs, and two- and three-level open Fermi structures. Similarly, EP4s
and closely related EP5s, which arise due to multiple symmetries, are
accompanied by exotic EP arcs and open Fermi structures. For each case, we
provide an explicit example. We also comment on the topological charge of these
EPs, and discuss similarities and differences between symmetry-protected
higher-order EPs and EP2s.
