On the singular Weinstein conjecture and the existence of escape orbits for b-Beltrami fields

Motivated by Poincar\'e's orbits going to infinity in the (restricted) three-body (see [26] and [6]), we investigate generic existence of heteroclinic-like a neighbourhood critical set $b$-contact form. This is done using singular counterpart [3] Etnyre--Ghrist's contact/Beltrami correspondence [9], genericity results concerning eigenfunctions Laplacian established Uhlenbeck [29]. Specifically, analyze $b$-Beltrami vector fields on $b$-manifolds dimension $3$ prove that for asymptotically exact $b$-metric they exhibit escape orbits. We also show symmetric field an flat $b$-manifold has generalized periodic orbit at least $4$ Generalized are trajectories whose $\alpha$- $\omega$-limit sets intersect surface. These first step towards proving Weinstein conjecture.