  A full amplitude analysis of $\Lambda_b^0 \to J/\psi p \pi^-$ decays is
performed with a data sample acquired with the LHCb detector from 7 and 8 TeV
$pp$ collisions, corresponding to an integrated luminosity of 3 fb$^{-1}$. A
significantly better description of the data is achieved when, in addition to
the previously observed nucleon excitations $N\to p\pi^-$, either the
$P_c(4380)^+$ and $P_c(4450)^+\to J/\psi p$ states, previously observed in
$\Lambda_b^0 \to J/\psi p K^-$ decays, or the $Z_c(4200)^-\to J/\psi \pi^-$
state, previously reported in $B^0 \to J/\psi K^+ \pi^-$ decays, or all three,
are included in the amplitude models. The data support a model containing all
three exotic states, with a significance of more than three standard
deviations. Within uncertainties, the data are consistent with the
$P_c(4380)^+$ and $P_c(4450)^+$ production rates expected from their previous
observation taking account of Cabibbo suppression.
