= Solution
The velocity divergence is $\partial_xu_x+\partial_yu_y=\alpha-\alpha=0$, satisfying the constant-density <incompressible flow> condition. Choose the <stream function> convention $(u_x,u_y)=(\psi_y,-\psi_x)$. Integration then gives
$$
\boxed{\psi(x,y)=\alpha xy-\frac\beta2(x^2+y^2)+C.}
$$
Indeed, $\psi_y=\alpha x-\beta y$ and $-\psi_x=\beta x-\alpha y$. <Streamlines> are contours of this function. The <vector field> is also compatible with steady inviscid momentum balance without body forces: $(\mathbf u\cdot\nabla)\mathbf u=(\alpha^2-\beta^2)(x,y)$ is balanced by <pressure> $p=p_0-\rho(\alpha^2-\beta^2)(x^2+y^2)/2$.
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