No Hilton-Milner type results for linear groups of degree two

A set of permutations $\mathcal{F}$ of a finite transitive permutation group $G\leq \operatorname{Sym}(\Omega)$ is \emph{intersecting} if any pair of elements of $\mathcal{F}$ agree on an element of $\Omega$. We say that $G$ has the \emph{EKR property} if an intersecting set of $G$ has size at most the order of a point stabilizer. Moreover, $G$ has the \emph{strict-EKR} property whenever $G$ has the EKR property and any intersecting set of maximum size is a coset of a point stabilizer of $G$. It is known that the permutation group $\operatorname{GL}_2(\mathbb{F}_q)$ acting on $\Omega_q := \mathbb{F}_q^2\setminus\{0\}$ has the EKR property, but does not have the strict-EKR property since the stabilizer of a hyperplane is a maximum intersecting set. In this paper, it is proved that the Hilton-Milner type result does not hold for $\operatorname{GL}_2(\mathbb{F}_q)$ acting on $\Omega_q$. Precisely, it is shown that a maximal intersecting set of $\operatorname{GL}_2(\mathbb{F}_q)$ is of maximum size. As a result, we prove the Complete Erd\H{o}s-Ko-Rado theorem for $\operatorname{GL}_2(\mathbb{F}_{q})$.