[Solution] (1) + (2) gives
$$\frac{(x+y)\left(1-\sqrt{x^{2}-y^{2}}\right)}{\sqrt{1-x^{2}+y^{2}}}=a+b$$
(1) - (2) gives
$$\frac{(x-y)\left(1+\sqrt{x^{2}-y^{2}}\right)}{\sqrt{1-x^{2}+y^{2}}}=a-b$$
(3) $\times$ (4) gives
$$x^{2}-y^{2}=a^{2}-b^{2} .$$

Substituting (5) into (3) and (4) respectively gives
$$\left\{\begin{array}{l} \frac{(x+y)\left(1-\sqrt{a^{2}-b^{2}}\right)}{\sqrt{1-a^{2}+b^{2}}}=a+b \\ \frac{(x-y)\left(1+\sqrt{a^{2}-b^{2}}\right)}{\sqrt{1-a^{2}+b^{2}}}=a-b . \end{array}\right.$$

Solving gives
$$\left\{\begin{array}{l} x=\frac{a+b \sqrt{a^{2}-b^{2}}}{\sqrt{1-a^{2}+b^{2}}} \\ y=\frac{b+a \sqrt{a^{2}-b^{2}}}{\sqrt{1-a^{2}+b^{2}}} \end{array}\right.$$

Upon verification, when $0 \leqslant a^{2}-b^{2}<1$, (8) is the solution to the original system of equations; in other cases, the original system of equations has no real solutions.