  Exceptional points are interesting physical phenomena in non-Hermitian
physics at which the eigenvalues are degenerate and the eigenvectors coalesce.
In this paper, we find that the universal feature of arbitrary non-Hermitian
two level systems with singularities is basis defectiveness rather than energy
degeneracy or state coalescence. This leads to the discovery of general
exceptional points (GEPs). For GEPs, more subtle structures (e.g., Bloch
peach), additional classification, and' 'hidden" quantum phase transitions are
explored. By using the topologically protected subspace from two edge states in
the non-Hermitian SSH model as an example, we illustrate the physical
properties of different types of GEPs.
