On the simplest sextic fields and related Thue equations

We consider the parametric family of sextic Thue equations $$x^6-2m{x}^{5}y-5(m+3)x^4{y}^2-20x^3{y}^3+5m{x}^2{y}^4+2(m+3)x{y}^5+y^6=\lambda$$ 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 $xy(x+y)(x-y)(x+2y)(2x+y)=0$.