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We exhibit a non-varying phenomenon for the counting problem of cylinders, weighted by their area, passing through two marked (regular) Weierstrass points of a translation surface in a hyperelliptic connected component   $\mathcal{H}^{hyp}(2g-2)$   or   $\mathcal{H}^{hyp}(g-1,g-1)$  ,   $g&gt; 1$  . As an application, we obtain the non-varying phenomenon for the counting problem of (weighted) periodic trajectories on the Ehrenfest wind-tree model, a billiard in the plane endowed with   $\mathbb{Z}^2$  -periodically located identical rectangular obstacles.

A Non-varying Phenomenon with an Application to the Wind-Tree Model