  The i-th eigenvalue of the Laplacian on a surface can be viewed as a
functional on the space of Riemannian metrics of fixed area. Extremal points of
these functionals correspond to surfaces admitting minimal isometric immersions
into spheres. Recently, critical metrics for the first eigenvalue were
classified on tori and on Klein bottles. The present paper is concerned with
extremal metrics for higher eigenvalues on these surfaces. We apply a classical
construction due to Lawson. The ranks of the extremal eigenvalues are obtained
for the bipolar surfaces $\tilde \tau_{r,k}$ of the corresponding Lawson's tori
or Klein bottles. Furthermore, we find explicitly the $S^1$-equivariant minimal
immersion of the bipolar surfaces into $S^4$ by the corresponding
eigenfunctions.
