  Using electromagnetic corrections previously calculated by means of a
potential model, we have made a phase-shift analysis of the $\pi^\pm p$
elastic-scattering data up to a pion laboratory kinetic energy of 100 MeV. The
hadronic interaction was assumed to be isospin invariant. We found that it was
possible to obtain self-consistent databases by removing very few measurements.
A pion-nucleon model was fitted to the elastic-scattering database obtained
after the removal of the outliers. The model-parameter values showed an
impressive stability when the database was subjected to different criteria for
the rejection of experiments. Our result for the pseudovector $\pi N N$
coupling constant (in the standard form) is $0.0733 \pm 0.0014$. The six
hadronic phase shifts up to 100 MeV are given in tabulated form. We also give
the values of the s-wave scattering lengths and the p-wave scattering volumes.
Big differences in the s-wave part of the interaction were observed when
comparing our hadronic phase shifts with those of the current GWU solution. We
demonstrate that the hadronic phase shifts obtained from the analysis of the
elastic-scattering data cannot reproduce the measurements of the $\pi^- p$
charge-exchange reaction, thus corroborating past evidence that the hadronic
interaction violates isospin invariance. Assuming the validity of the result
obtained within the framework of chiral perturbation theory, that the mass
difference between the $u$- and the $d$-quark has only a very small effect on
the isospin invariance of the purely hadronic interaction, the
isospin-invariance violation revealed by the data must arise from the fact that
we are dealing with a hadronic interaction which still contains residual
effects of electromagnetic origin.
