The singular Weinstein conjecture for b-contact manifolds

Let $(M,Z,\alpha)$ be a compact $b^m$-contact manifold, with critical set $Z$. A singular periodic orbit is an integral curve of the Reeb $b$-vector field in $M\setminus Z$ whose limits as $t\to\pm\infty$ are points of $Z$. The singular Weinstein conjecture. There exists at least one singular periodic orbit or one periodic orbit outside of the critical set. The paper constructs counterexamples to this claim, so it is refuted.