  We introduce the notion of a generalised symmetry M of a hamiltonian H. It is
a symmetry which has been broken in a very specific manner, involving ladder
operators R and R*. In Theorem 1 these generalised symmetries are characterised
in terms of repeated commutators of H with M. Breaking supersymmetry by adding
a term linear in the supercharges is discussed as a motivating example. The
complex parameter gamma which appears in the definition of a generalised
symmetry is necessarily real when the spectrum of M is discrete. Theorem 2
shows that gamma must also be real when the spectrum of H is fully discrete and
R and R* are bounded operators. Any generalised symmetry induces a partitioning
of the spectrum of H in what we call M-multiplets. The hydrogen atom in the
presence of a symmetry breaking external field is discussed as an example. The
notion of stability of eigenvectors of H relative to the generalised symmetry M
is discussed. A characterisation of stable eigenvectors is given in Theorem 3.
