  We perform a lattice study of charmonium-like mesons with $J^{PC}=1^{++}$ and
three quark contents $\bar cc \bar du$, $\bar cc(\bar uu+\bar dd)$ and $\bar cc
\bar ss$, where the later two can mix with $\bar cc$. This simulation with
$N_f=2$ and $m_\pi=266$ MeV aims at the possible signatures of four-quark
exotic states. We utilize a large basis of $\bar cc$, two-meson and
diquark-antidiquark interpolating fields, with diquarks in both anti-triplet
and sextet color representations. A lattice candidate for X(3872) with I=0 is
observed very close to the experimental state only if both $\bar cc$ and $D\bar
D^*$ interpolators are included; the candidate is not found if
diquark-antidiquark and $D\bar D^*$ are used in the absence of $\bar cc$. No
candidate for neutral or charged X(3872), or any other exotic candidates are
found in the I=1 channel. We also do not find signatures of exotic $\bar cc\bar
ss$ candidates below 4.3 GeV, such as Y(4140). Possible physics and methodology
related reasons for that are discussed. Along the way, we present the
diquark-antidiquark operators as linear combinations of the two-meson operators
via the Fierz transformations.
