Jacobian elliptic fibrations on K3s with a non-symplectic automorphism of order 3

Let $X$ be a K3 surface admitting a non-symplectic automorphism $\sigma$ of order 3. Building on work by Garbagnati and Salgado, we classify the Jacobian elliptic fibrations on $X$ with respect to the action of $\sigma$ on their fibers. If the fiber class of a Jacobian elliptic fibration on $NS\left(X\right)$ is fixed by $\sigma$, we determine the possible configurations of its singular fibers and present the equation for its generic fiber. When the Picard number of $X$ is at least 12 and $\sigma$ acts trivially on $NS\left(X\right)$, we apply the Kneser–Nishiyama method to find its Jacobian elliptic fibrations up to $J_2$-equivalence. We use our method to classify them with respect to any non-symplectic automorphism of order 3 in $Aut\left(X\right)$.