= Solution
The <stream function> gives $u=2\alpha xy$ and $v=-\alpha y^2$. Its scalar <vorticity> is
$$
v_x-u_y=-2\alpha x.
$$
For $\alpha\ne0$ this does not vanish on any nonempty open region of the plane. A <holomorphic> <complex potential of a two-dimensional potential flow> would produce an <irrotational flow> there. Equivalently, its imaginary part would be a <harmonic function>, whereas $\Delta\psi=2\alpha x$. Thus \b[no complex potential exists on any nonempty open region for this flow]; vanishing <vorticity> only along the vertical line does not provide a two-dimensional domain.
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