We prove the following conjecture recently formulated by Jakobson,
Nadirashvili and Polterovich \cite{JNP}: on the Klein bottle K, the
metric of revolution g0=1+8cos2v9+(1+8cos2v)2(du2+1+8cos2vdv2),0≤u<2π, 0≤v<π, is the
\emph{unique} extremal metric of the first eigenvalue of the Laplacian viewed
as a functional on the space of all Riemannian metrics of given area. The proof
leads us to study a Hamiltonian dynamical system which turns out to be
completely integrable by quadratures