  We generalise the form factor bootstrap approach to integrable field theories
with U(1) symmetry to derive matrix elements of composite branch-point twist
fields associated with symmetry resolved entanglement entropies. The bootstrap
equations are solved for the free massive Dirac and complex boson theories,
which are the simplest theories with U(1) symmetry. We present the exact and
complete solution for the bootstrap, including vacuum expectation values and
form factors involving any type and arbitrarily number of particles. The
non-trivial solutions are carefully cross-checked by performing various limits
and by the application of the Delta-theorem. An alternative and compact
determination of the novel form factors is also presented. Based on the form
factors of the U(1) composite branch-point twist fields, we re-derive earlier
results showing entanglement equipartition for an interval in the ground state
of the two models.
