= Solution
A continuously differentiable <conservative vector field> has zero <curl>. Here
$$
\nabla\times\mathbf F=
\left(\frac{\partial F_z}{\partial y}-\frac{\partial F_y}{\partial z},\frac{\partial F_x}{\partial z}-\frac{\partial F_z}{\partial x},\frac{\partial F_y}{\partial x}-\frac{\partial F_x}{\partial y}\right)
=\bigl(\beta(2y-1),0,0\bigr).
$$
This vanishes everywhere if and only if $\boxed{\beta=0}$. Since the field is smooth on the <simply connected> domain $\mathbb R^3$, zero <curl> is also sufficient; the explicit <scalar potential> in the next part confirms it.
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