  We consider the parametric family of sextic Thue equations \[
x^6-2mx^5y-5(m+3)x^4y^2-20x^3y^3+5mx^2y^4+2(m+3)xy^5+y^6=\lambda \] where
$m\in\mathbb{Z}$ is an integer and $\lambda$ is a divisor of $27(m^2+3m+9)$. We
show that the only solutions to the equations are the trivial ones with
$xy(x+y)(x-y)(x+2y)(2x+y)=0$.
