  A new approach is proposed for study structure and properties of the total
squared mean curvature $W$ of surfaces in ${\bf R}^3$. It is based on the
generalized Weierstrass formulae for inducing surfaces. The quantity $W$
(Willmore functional) is shown to be invariant under the modified
Novikov--Veselov hierarchy of integrable flows. The $1+1$--dimensional case
and, in particular, Willmore tori of revolution, are studied in details. The
Willmore conjecture is proved for the mKDV--invariant Willmore tori.
