  We consider an initial data set having a continuous symmetry and a marginally
outer trapped surface (MOTS) that is not preserved by this symmetry. We show
that such a MOTS is unstable except in an exceptional case. In non-rotating
cases we provide a Courant-type lower bound on the number of unstable
eigenvalues. These results are then used to prove the instability of a large
class of exotic MOTSs that were recently observed in the Schwarzschild
spacetime. We also discuss the implications for the apparent horizon in data
sets with translational symmetry.
