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Let $𝒜$ be the real vector space of Abelian integrals $$I\left(h\right)=\int {\int }_{\{H\le h\}}R(x,y)dx\wedge dy,h\in [0,\tilde{h}]$$ where $H(x,y)=(x^2+y^2)/2+...$ is a fixed real polynomial, $R(x,y)$ is an arbitrary real polynomial and $\{H\le h\}$, $h\in [0,\tilde{h}]$, is the interior of the oval of $H$ which surrounds the origin and tends to it as $h\to 0$. We prove that if $H(x,y)$ is a semiweighted homogeneous polynomial with only Morse critical points, then $𝒜$ is a free finitely generated module over the ring of real polynomials $ℝ\left[h\right]$, and compute its rank. We find the generators of $𝒜$ in the case when $H$ is an arbitrary cubic...