We prove the following conjecture recently formulated by Jakobson, Nadirashvili, and Polterovich (see [15, Conjecture 1.5, page 383]). On the Klein bottle $\mathbb{K}$ , the metric of revolution $g_0={9+ (1+8\cos^2v)^2\over 1+8\cos^2v} \Big(du^2 + {dv^2\over 1+8\cos^2v}\Big),$ $0\le u\textless\pi/2$ , $0\le v\textless\pi$ , is the 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 that turns out to be completely integrable by quadratures