Polynomial Solutions of Equivariant Polynomial Abel Differential Equations

Let a(x) be non-constant and let b(j)(x), for j = 0, 1, 2, 3, be real or complex polynomials in the variable x. Then the real or complex equivariant polynomial Abel differential equation a(x)(y) over dot = b(1)(x) y + b(3)(x) y(3), with b(3)(x) not equal 0, and the real or complex polynomial equivariant polynomial Abel differential equation of the second kind a(x)y(y) over dot = b(0)(x) + b(2)(x) y(2), with b(2)(x) not equal 0, have at most 7 polynomial solutions. Moreover, there exist equations of this type having this maximum number of polynomial solutions.