The image in $H^2(Q^3;\mathbb R)$ of the set of presymplectic forms with a prescribed kernel

A new invariant $\Omega$ of a 1-distribution $\mathscr I$ on a closed 3-dimensional manifold $Q^3$ is defined as the domain in the second cohomology group $H^2(Q^3;\mathbb R)$ generated by the restrictions to $Q^3=Q^3\times \{0\}$ of all symplectic forms $\omega$ on $Q^3\times \mathbb R$ such that the kernel of the restriction $\omega \big |_{Q^3}$ is the 1-distribution $\mathscr I$ (that is, $\mathscr I$ is the characteristic distribution of this restriction). This invariant is calculated in the cases when the distribution $\mathscr I$ is non-integrable, Bott non-resonance integrable, and resonance integrable.