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