We prove that for $a\in {\mathbb R}\smallsetminus \{0\}$, the polynomial endomorphism $F:{\mathbb C}^2\to {\mathbb C}^2$ defined by \[F\left(\begin{array}{c}x\\y\end{array}\right) = \left(\begin{array}{c}x+y^2+ax(xy)\\y+x^2+ay(x-y)\end{array}\right)\] has infinitely many distinct invariant Fatou components in which orbits converge to the origin.