  We study the structure of the eigenstates and the dynamics of a system that
undergoes an excited state quantum phase transition (ESQPT). The analysis is
performed for two-level pairing models characterized by a U(n+1) algebraic
structure. They exhibit a second order phase transition between two limiting
dynamical symmetries represented by the U(n) and SO(n+1) subalgebras. They are,
or can be mapped onto, models of interacting bosons. We show that the
eigenstates with energies very close to the ESQPT critical point, E_{ESQPT},
are highly localized in the U(n)-basis. Consequently, the dynamics of a system
initially prepared in a U(n)-basis vector with energy close to E_{ESQPT} may be
extremely slow. Signatures of an ESQPT can therefore be found in the structures
of the eigenstates and in the speed of the system evolution after a sudden
quench. Our findings can be tested experimentally with trapped ions.
