Center problem for quartic differential systems with an affine invariant straight line of maximal multiplicity

In this paper the quartic differential systems with a center-focus critical point and non-degenerate infinity are examined. We show that in this family the maximal multiplicity of an affine invariant straight line is six. Modulo the affine transformation and time rescaling the classes of systems with an affine invariant straight line of multiplicity two, three,..., six are determined. In the cases when the quartic systems has an affine invariant straight line of maximal multiplicity the problem of the center is solved.