  We formulate the final state interaction of the $a_1(1260)$ resonance decay
in a manifestly three-body unitary parameterization and fit it to the
$a_1(1260)$ lineshape measured by the ALEPH experiment. Dalitz plots calculated
from this fit are presented. The work demonstrates the feasibility to
numerically solve a previously derived amplitude and its generalization to
isobars with spin and coupled channels. The model can also be applied to other
meson decays and modified for the finite-volume problem as it arises in lattice
QCD due to its manifest unitarity.
