  By looking at the pseudoscalar-vector meson spectra in the $\bar{B}\to J/\psi
\rho\bar{K}$ and $\bar{B}\to J/\psi \bar{K}^*\pi$ weak decays, we theoretically
investigate the double-pole structure of the $K_1(1270)$ resonance by using the
Chiral Unitary approach to account for the final state interactions between the
pseudoscalar and vector mesons. The $K_1(1270)$ resonance appears as
dynamically generated through these interactions in coupled channels and
influence the shape of the invariant mass distributions under consideration. We
show how those shapes are affected by the $K_1(1270)$ double-pole structure to
confront the results from our model with future experiments that might
investigate the $PV$ spectra in these decays.
