As stated here, Uniqueness of Helmholtz decomposition? , the solution of the Helmholtz decomposition is not unique.
Suppose that, for given vector field $\mathbf F$ with $\nabla \cdot \mathbf F =0$, I have a solution of its Helmholtz decomposition: the pair $\phi$ and $\mathbf A$. They are such that: $$\mathbf F = -\nabla \phi + \nabla\times\mathbf A.$$
What kind of transformation can I apply to $\phi$ and $\mathbf A$ to find another pair $\phi_1$ and $\mathbf A_1$ such that $$\nabla \phi_1 = 0$$ and $$\mathbf F=\nabla \times \mathbf A_1 $$?