1. Solve the following equations in the set of complex numbers:
(1) $2 z - \mathrm{i} \bar{z} = 1$;
(2) $\bar{z} - \lambda z = \omega$ (constants $\lambda, \omega \in \mathbf{C}$, and $|\lambda| \neq 1$).
(1) Taking the conjugate of both sides of the original equation yields $2 \bar{z}+\mathrm{i} z=1$,

Combining the two equations $\left\{\begin{array}{l}2 z-i \bar{z}=1, \\ 2 \bar{z}+\mathrm{i} z=1 .\end{array}\right.$
Solving this system of equations gives $z=\frac{2}{3}+\frac{1}{3} \mathrm{i}$.
(2) Taking the conjugate of both sides of the original equation yields $z-\bar{\lambda} \cdot \bar{z}=\bar{\omega}$,

Combining the two equations $\left\{\begin{array}{l}\bar{z}-\lambda z=\omega, \\ z-\bar{\lambda} \cdot \bar{z}=\bar{\omega} .\end{array}\right.$ we get $(1-\lambda \cdot \bar{\lambda}) \cdot z=\bar{\lambda} \cdot \omega+\bar{\omega}$.
Since $|\lambda| \neq 1$, then $1-\lambda \cdot \bar{\lambda} \neq 0$, hence $z=\frac{\bar{\lambda} \cdot \omega+\bar{\omega}}{1-\lambda \cdot \bar{\lambda}}$.