  We consider small polynomial deformations of integrable systems of the form
$dF=0$, $F\in\mathbb{C}[x,y]$ and the first nonzero term $M_\mu$ of the
displacement function $\Delta(t,\epsilon)=\sum_{i=\mu}M_i(t)\epsilon^i$ along a
cycle $\gamma(t)\in F^{-1}(t)$. It is known that $M_\mu$ is an iterated
integral of length at most $\mu$. The bound $\mu$ depends on the deformation of
$dF$.
  In this paper we give a universal bound for the length of the iterated
integral expressing the first nonzero term $M_\mu$ depending only on the
topology of the unperturbed system $dF=0$. The result generalizes the result of
Gavrilov and Iliev providing a sufficient condition for $M_\mu$ to be given by
an abelian integral i.e. by an iterated integral of length $1$. We conjecture
that our bound is optimal.
