3. Solve the system of equations $\left\{\begin{array}{l}x-y+z-w=2, \\ x^{2}-y^{2}+z^{2}-w^{2}=6, \\ x^{3}-y^{3}+z^{3}-w^{3}=20, \\ x^{4}-y^{4}+z^{4}-w^{4}=66 .\end{array}\right.$
Let $u=x+z, v=x z, s=y+w, t=y w$,
then $x^{2}+z^{2}=u^{2}-2 v, x^{3}+z^{3}=u^{3}-3 u v, x^{4}+z^{4}=u^{4}-4 u^{2} v+2 v^{2}$.
Thus, the original system of equations can be transformed into $\left\{\begin{array}{l}u-s=2, \\ u^{2}-s^{2}-2 v+2 t=6, \\ u^{3}-s^{3}-3 u v+3 s t=20, \\ u^{4}-s^{4}-4 u^{2} v+4 s^{2} t+2 v^{2}-2 t^{2}=66 .\end{array}\right.$
From (1), we get $u=s+2, u^{2}=s^{2}+4 s+4$, substituting into (2) yields $t-v=1-2 s$. (5) $u^{3}=s^{3}+6 s^{2}+12 s+8$, substituting into (3) yields
$$
\begin{array}{l}
6 s^{2}+12 s+8-3(s+2) v+3 s t=20 \Rightarrow 6 s^{2}+12 s+3 s(t-v)-6 v=12 \\
\Rightarrow 2 s^{2}+4 s+s(1-2 s)-2 v=4 \Rightarrow 2 v=5 s-4 .
\end{array}
$$

Combining (5) and (6), we get $v=\frac{5}{2} s-2, t=\frac{1}{2} s-1 \Rightarrow 2 v^{2}-2 t^{2}=12 s^{2}-18 s+6$. $u^{4}=s^{4}+8 s^{3}+24 s^{2}+32 s+16$, substituting into (4) yields
$$
\begin{array}{l}
8 s^{3}+24 s^{2}+32 s+16-2\left(s^{2}+4 s+4\right)(5 s-4)+2 s^{2}(s-2)+12 s^{2}-18 s+6 \\
=8 s^{3}+36 s^{2}+14 s-2\left(10 s^{3}+16 s^{2}+4 s-16\right)+\left(2 s^{3}-4 s^{2}\right)+22 \\
=6 s+54=66 \Rightarrow s=2 \Rightarrow u=4, v=3, t=0 .
\end{array}
$$

Thus, $x, z$ are the roots of the equation $p^{2}-4 p+3=0 \Rightarrow x, z=1,3$; $y, w$ are the roots of the equation $q^{2}-2 q=0 \Rightarrow y, w=0,2$. In summary, the solutions to the original system of equations are four groups: $(1,0,3,2),(1,2,3,0),(3,0,1,2),(3,2,1,0)$.