Greatest least eigenvalue of the Laplacian on the Klein bottle

We prove the following conjecture recently formulated by Jakobson, Nadirashvili and Polterovich \cite{JNP}: For any Riemannian metric $g$ on the Klein bottle $\mathbb{K}$ one has $$\lambda \mathrm{\_}1(\mathbb{K},g)A(\mathbb{K},g)\le 12\pi E(2\sqrt{2}\mathrm{/}3),$$ where $\lambda \mathrm{\_}1(\mathbb{K},g)$ and $A(\mathbb{K},g)$ stand for the least positive eigenvalue of the Laplacian and the area of $(\mathbb{K},g)$, respectively, and $E$ is the complete elliptic integral of the second kind. Moreover, the equality is uniquely achieved, up to dilatations, by the metric $$g\mathrm{\_}0=\frac{9+(1+8{\cos }^{2}v{)}^2}{1+8{\cos }^{2}v}(d{u}^2+\frac{d{v}^2}{1+8{\cos }^{2}v}),$$ with $0\le u,v<\pi$. The proof of this theorem leads us to study a Hamiltonian dynamical system which turns out to be completely integrable by quadratures.