4. Let $z, \omega, \lambda$ be complex numbers, $|\lambda| \neq 1$, and consider the equation $z-\overline{\lambda z}=\bar{\omega}$ with respect to $z$. There are four conclusions: (I) $z=$ $\frac{\bar{\lambda} \omega+\bar{\omega}}{1-|\lambda|^{2}}$ is a solution to this equation; (II) This equation has only one solution; (III) This equation has two solutions; (IV) This equation has infinitely many solutions. Then
A. Only I and II are correct
B. Only I and III are correct
C. Only I and IV are correct
D. None of (A), (B), (C) are correct
4. $\left\{\begin{array}{l}\bar{z}-\lambda z=\omega \\ z-\overline{\lambda z}=\bar{\omega}\end{array} \Rightarrow\left\{\begin{array}{l}\overline{\lambda z}-\lambda \bar{\mu} z=\bar{\lambda} \omega \\ z-\overline{\lambda z}=\bar{\omega}\end{array} \Rightarrow\right.\right.$ Therefore, $z(1-\bar{\mu})=\bar{\omega}+\bar{\lambda} \omega$ so $z=\frac{\bar{\omega}+\omega \bar{\lambda}}{1-|\lambda|^{2}}$.