  We introduce a near-threshold parameterization that is more general than the
effective-range expansion up to and including the effective-range because it
can also handle with a near-threshold zero in the $D^0\bar{D}^{*0}$ $S$-wave.
In terms of it we analyze the CDF data on inclusive $p\bar{p}$ scattering to
$J/\psi \pi^+\pi^-$, and the Belle and BaBar data on $B$ decays to $K\, J/\psi
\pi^+\pi^-$ and $K D\bar{D}^{*0}$ around the $D^0\bar{D}^{*0}$ threshold. It is
shown that data can be reproduced with similar quality for the $X(3872)$ being
a bound {\it and/or} a virtual state. We also find that the $X(3872)$ might be
a higher-order virtual-state pole (double or triplet pole), in the limit in
which the small $D^{*0}$ width vanishes. Once the latter is restored the
corrections to the pole position are non-analytic and much bigger than the
$D^{*0}$ width itself. The $X(3872)$ compositeness coefficient in
$D^0\bar{D}^{*0}$ ranges from nearly 0 up to 1 in the different scenarios.
