  The paper deals with the {\it infinitesimal Hilbert 16th problem}: to find an
upper estimate of the number of zeros of an Abelian integral regarded as a
function of a parameter. In more details, consider a real polynomial $ H$ of
degree $ n+1 $ in the plane, and a continuous family of ovals $\gamma_t$
(compact components of level curves $ H = t$) of this polynomial. Consider a
polynomial 1-form $\omega$ with coefficients of degree at most $n.$ Let I(t) =
\int_{\gamma_t} \omega. \label{I}
  The problem is to give an upper estimate of the number of zeros of this
integral. We solve a {\it restricted version} of this problem. Namely, the form
$ \omega $ is {\it arbitrary,}, and the polynomial $ H$, though having an
arbitrary degree, is not too close to the hypersurface of degenerate (non
ultra-Morse) polynomials. We hope that the solution of the restricted version
of the problem is a step to the solution of the complete (nonrestricted)
version.
