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We introduce two square-tiled surfaces, one with $8$ squares inside $\Omega\mathcal{M}_3(2,2)$, and the other with $9$ squares inside $\Omega\mathcal{M}_4(3,3)$, respectively. In these examples, the dimensions of theisotropic subspaces (in absolute homology) generated by the waist curves of themaximal cylinders in any fixed rational direction are $2$ and $3$ respectively.Hence, a geometrical criterion of G. Forni for the non-uniform hyperbolicity ofKontsevich-Zorich (KZ) cocycle can not be applied to these examples.Nevertheless, we prove that there are no vanishing exponents and the spectrumis simple for these two square-tiled surfaces. In particular, the non-vanishingof exponents of KZ cocycle for a regular measure doesn't imply that the supportof this measure contains a completely periodic surface whose waist curves ofmaximal cylinders generates a Lagrangian subspace in its absolute homology.

Un contre-exemple \`a la r\'eciproque du crit\`ere de Forni pour la positivit\'e des exposants de Lyapunov du cocycle de Kontsevich-Zorich