  We consider a class of foliations on the complex projective plane that are
determined by a quadratic vector field in a fixed affine neighborhood. Such
foliations, as a rule, have an invariant line at infinity. Two foliations with
singularities on $\mathbb C P^2$ are topologically equivalent provided that
there exists a homeomorphism of the projective plane onto itself that preserves
orientation both on the leaves and in $\mathbb C P^2$ and brings the leaves of
the first foliation to that of the second one. We prove that a generic
foliation of this class may be topologically equivalent to but a finite number
of foliations of the same class, modulo affine equivalence. This property is
called \emph{total rigidity}. Recent result of Lins Neto implies that the
finite number above does not exceed 240. This is the first of the two closely
related papers. It deals with the rigidity properties of quadratic foliations,
whilst the second one studies the foliations of higher degree.
