  We investigate the existence of higher order exceptional points (EPs) in
non-Hermitian systems, and show that $\mu$-fold EPs are stable in $\mu-1$
dimensions in the presence of anti-unitary symmetries that are local in
parameter space, such as e.g. PT or CP symmetries. This implies in particular
that 3-fold and 4-fold symmetry-protected EPs are stable respectively in 2 and
3 dimensions. The stability of such exceptional points is expressed in terms of
the homotopy properties of a ''resultant vector'' that we introduce. Our
framework also allows us to rephrase the previously proposed $\mathbb{Z}_2$
index of PT and CP symmetric gapped phases beyond the realm of two-band models.
We apply this general formalism to a frictional shallow water model that is
found to exhibit 3-fold exceptional points associated with topological numbers
$\pm1$. For this model, we also show different non-Hermitian topological
transitions associated with these exceptional points, such as their merging and
a transition to a regime where propagation becomes forbidden.
