  Extremal spectral properties of Lawson tau-surfaces are investigated. The
Lawson tau-surfaces form a two-parametric family of tori or Klein bottles
minimally immersed in the standard unitary three-dimensional sphere. A Lawson
tau-surface carries an extremal metric for some eigenvalue of the
Laplace-Beltrami operator. Using theory of the Lam\'e equation we find
explicitly these extremal eigenvalues.
