  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 $$\lambda\_1 (\mathbb{K}, g) A (\mathbb{K},
g)\le 12 \pi E(2\sqrt 2/3),$$ where $\lambda\_1(\mathbb{K},g)$ 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 $$g\_0= {9+ (1+8\cos ^2v)^2\over 1+8\cos^2v} (du^2 +
{dv^2\over 1+8\cos ^2v}),$$ 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.
