There exists a local 1-form $\theta$ on a neighborhood of $\mathcal{P}$ in $\mathbb{L}$ with the following properties:

1. (1)

   $\theta=O^{\prime}(s^{2})$,

2. (2)

   $\theta$ is smooth away from $\pi^{-1}(P)$,

3. (3)

   $\mathcal{L}_{\partial_{t}}\theta=0$,

4. (4)

   $\partial_{t}\lrcorner\theta=0$,

5. (5)

   $d(\Theta_{1}+\theta)=\Upsilon$.