Linear estimate for the number of zeros of Abelian integrals

We prove a linear in $\mathrm{deg}\omega$ upper bound on the number of real zeros of the Abelian integral $I(t)={\int }_{\delta (t)}\omega$, where $\delta (t)\subset {\mathbb{R}}^2$ is the real oval $x^{2}y(1-x-y)=t$ and $\omega$ is a one-form with polynomial coefficients.