Let $x=a+b, y=a-b$. Then $x+y=2a, xy=a^{2}-b^{2}$.
Thus, the original system of equations becomes
$$\left\{\begin{array} { l } { a ^ { 2 } - b ^ { 2 } + 2 a = 1 , } \\ { 2 a ^ { 2 } + 2 b ^ { 2 } + ( a ^ { 2 } - b ^ { 2 } ) ^ { 2 } = 1 7 } \end{array} \Rightarrow \left\{\begin{array}{l} a=\frac{3}{2}, \\ b= \pm \frac{\sqrt{17}}{2} \end{array}\right.\right.$$

Therefore, the solutions to the original system of equations are
$$\left\{\begin{array} { l } { x _ { 1 } = \frac { 3 + \sqrt { 1 7 } } { 2 } , } \\ { y _ { 1 } = \frac { 3 - \sqrt { 1 7 } } { 2 } ; } \end{array} \quad \left\{\begin{array}{l} x_{2}=\frac{3-\sqrt{17}}{2}, \\ y_{2}=\frac{3+\sqrt{17}}{2} . \end{array}\right.\right.$$