Consider the equation

$w-\bar{w} z=k(1-z), k \in R$

Clearly $z \neq 1$ and $\frac{w-\bar{w} z}{1-z}$ is purely real

$\therefore \frac{\overline{w-\bar{w} z}}{1-z}=\frac{w-\bar{w} z}{1-z}$

$\Rightarrow \frac{\bar{w}-w \bar{z}}{1-\bar{z}}=\frac{w-\bar{w} z}{1-z}$

$\Rightarrow \bar{w}-\bar{w} z-w \bar{z}+w \overline{z z}$

$=w-w \bar{z}-\bar{w} z+\bar{w} z \bar{z}$

$\Rightarrow \bar{w}+w|z|^{2}=w+\bar{w}|z|^{2}$

$ \Rightarrow (w – \bar w)(|z{|^2}) = w – \bar w$

$\Rightarrow|z|^{2}=1$ $ \quad(\because \operatorname{Im} w \neq 0)$

$\Rightarrow|z|=1$ and $z \neq 1$

$\therefore$ The required set is $\{z:|z|=1, z \neq 1\}$