Let $i:T^{2}\rightarrow S^{1}\times S^{2}$ be a embedding map and $i_{*}:\pi(T^{2})\rightarrow \pi(S^{1}\times S^{2})$ be the homomorphism induced by $i$. If $i(T^{2})$ is separable in $S^{1}\times S^{2}$ and $Im i_{*}=0$, then $$S^{1}\times S^{2}-i(T^{2})\cong K^{c}\cup (S^{1}\times D^{2})\sharp (S^{1}\times D^{2})$$ or

$$S^{1}\times S^{2}-i(T^{2})\cong (S^{1}\times D^{2})\cup(S^{1}\times D^{2})\sharp K^{c}$$

Here $S^{1}\times D^{2}$ is an open solid torus and $K^{c}$ compliment knot on $S^{3}$.