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Let $u$ be an inner function and $K_u^2$ be the corresponding model space. For an inner function $v$, the subspace $v{H}^2$ is an invariant subspace of the unilateral shift operator on $H^2$. In this article, using the structure of a Toeplitz kernel $\ker T_{\overline{u}v}$, we study the intersection $K_u^2\cap v{H}^2$ by properties of inner functions $u$ and $v$ $(v\ne u)$. If $K_u^2\cap v{H}^2\ne \left\{0\right\}$, then there exists a triple $(B,b,g)$ such that $$\overline{u}v=\frac{\lambda b\overline{B{O}_g}}{g},$$ where the triple $(B,b,g)$ means that $B$ and $b$ are Blaschke products, $g$ is an invertible function in $H^{\infty }$, $O_g$ denotes the outer factor of $g$, and $\lambda$...