  Many mesons with properties incompatible with a $\bar cc$ structure have
already been discovered, e.g. the $Z_c$ mesons with isospin 1. We investigate
the spectrum of exotic charmonium-like mesons using lattice QCD. The focus is
on $\bar cc \bar qq$ states with $J^{PC}=1^{+\pm}$ and isospin 1. This is the
first study of four-quark states with these quantum numbers, a non-zero total
momentum and two different lattice volumes. We extract the energy levels and
determine the scattering length for $D\bar D^*$ scattering close to the
threshold using L\"uscher's formalism. Our preliminary results show that the
energy shifts for eigenstates dominated by $D\bar{D}^*$ are very small in the
$1^{++}$ channel and consistent with zero in the $1^{+-}$ channel.
