Isospin properties of the X state near the D {\bar D}^{*} threshold

The $D {\bar D}^*$ scattering amplitude and the production of the final states $\pi^+ \pi^- J/\psi$ and $\pi^+ \pi^- \pi^0 J/\psi$ near the $D^0 {\bar D}^{*0}$ threshold are discussed following the recent suggestion that the observed peaks X(3872) and X(3875) in the decays $B \to X K$ are due to a virtual state $X$ in the $D^0 {\bar D}^{*0}$ channel. The strong interaction is treated using the small interaction radius approximation. It is shown that the mass difference between the charged and neutral $D^{(*)}$ mesons results in a distinctive behavior of the relevant isotopic amplitudes. In particular, the shape of the peak in the $\pi^+ \pi^- J/\psi$ channel should be significantly narrower than in the $\pi^+ \pi^- \pi^0 J/\psi$ channel, which property can be used for an experimental test of the virtual state hypothesis.