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Let $P(x)$ be a real polynomial of degree $2g+1$, $H=y^2+P(x)$ and$\delta(h)$ be an oval contained in the level set $\{H=h\}$. We study completeAbelian integrals of the form $$I(h)=\int_{\delta(h)} \frac{(\alpha_0+\alpha_1x+... + \alpha_{g-1}x^{g-1})dx}{y}, h\in \Sigma,$$ where $\alpha_i$ are realand $\Sigma\subset \R$ is a maximal open interval on which a continuous familyof ovals $\{\delta(h)\}$ exists. We show that the $g$-dimensional real vectorspace of these integrals is not Chebyshev in general: for any $g>1$, there arehyperelliptic Hamiltonians $H$ and continuous families of ovals$\delta(h)\subset\{H=h\}$, $h\in\Sigma$, such that the Abelian integral $I(h)$can have at least $[\frac32g]-1$ zeros in $\Sigma$. Our main result is Theorem\ref{main} in which we show that when $g=2$, exceptional families of ovals$\{\delta(h)\}$ exist, such that the corresponding vector space is stillChebyshev.

Complete hyperelliptic integrals of the first kind and their non-oscillation