There are also examples that are characterized dynamically. By Viterbo, every contact-type hypersurface of $\mathbb{R}^{2n}$ carries a closed Reeb orbit, i.e. a closed leaf of the characteristic line bundle. (Actually, Hofer and Zehnder show this is true for any stable hypersurface.) There are examples of autonomous Hamiltonians on $\mathbb{R}^{2n}$ that fail to have any closed orbits on a specific level set, thus providing examples of hypersurfaces that fail to be stable. These are due to Ginzburg and Gurel.

There are also examples that are characterized dynamically. By Viterbo, every compact, contact-type hypersurface of $\mathbb{R}^{2n}$ carries a closed Reeb orbit, i.e. a closed leaf of the characteristic line bundle. (Actually, Hofer and Zehnder show this is true for any stable hypersurface.) There are examples of autonomous Hamiltonians on $\mathbb{R}^{2n}$ that fail to have any closed orbits on a specific level set, thus providing examples of hypersurfaces that fail to be stable. These are due to Ginzburg and Gurel.