$$\left(
\begin{array}[c]{cc}
    E-m & -\vec p\cdot \vec \sigma\\
    \vec p \cdot \vec \sigma & - E - m
\end{array}\right)
\left(
\begin{array}[c]{c}
    \xi\\
    \eta
\end{array}\right)=0 \hfil\Leftrightarrow\hfil \begin{aligned}
    (E-m) \xi &= \vec p \cdot \vec \sigma \eta\\
    (E+m) \eta &= \vec p \cdot \vec \sigma \xi
\end{aligned}$$