Let
P
(
x
)
P(x)
be a real polynomial of degree
2
g
+
1
2g+1
,
H
=
y
2
+
P
(
x
)
H=y^2+P(x)
and
δ
(
h
)
\delta (h)
be an oval contained in the level set
{
H
=
h
}
\{H=h\}
. We study complete Abelian integrals of the form
\[
I
(
h
)
=
∫
δ
(
h
)
(
α
0
+
α
1
x
+
…
+
α
g
−
1
x
g
−
1
)
d
x
y
,
h
∈
Σ
,
I(h)=\int _{\delta (h)} \frac {(\alpha _0+\alpha _1 x+\ldots + \alpha _{g-1}x^{g-1})dx}{y}, \;\;h\in \Sigma ,
\]
where
α
i
\alpha _i
are real and
Σ
⊂
R
\Sigma \subset \mathbb {R}
is a maximal open interval on which a continuous family of ovals
{
δ
(
h
)
}
\{\delta (h)\}
exists. We show that the
g
g
-dimensional real vector space of these integrals is not Chebyshev in general: for any
g
>
1
g>1
, there are hyperelliptic Hamiltonians
H
H
and continuous families of ovals
δ
(
h
)
⊂
{
H
=
h
}
\delta (h)\subset \{H=h\}
,
h
∈
Σ
h\in \Sigma
, such that the Abelian integral
I
(
h
)
I(h)
can have at least
[
3
2
g
]
−
1
[\frac 32g]-1
zeros in
Σ
\Sigma
. Our main result is Theorem 1 in which we show that when
g
=
2
g=2
, exceptional families of ovals
{
δ
(
h
)
}
\{\delta (h)\}
exist, such that the corresponding vector space is still Chebyshev.