I am looking for $F: \mathbb{R}^2 \to \mathbb{R}^2$ such that $$\nabla F\cdot F = (1+ xy) \binom{x}{y},$$ where for $F = \binom{f_1}{f_2},$ we denote $\nabla F = \binom{\nabla f_1^T}{\nabla f_2^T}.$
Note that for $G(x,y) = \frac{1}{\sqrt{2}}\binom{y^2}{x^2},$ we have $\nabla G \cdot G = xy\binom{x}{y}$ and for $H(x,y) = \binom{x}{y},$ we have $\nabla H \cdot H = H$ but for $F = G + H,$ $$\nabla F\cdot F \neq (1+ xy) \binom{x}{y},$$ since the above problem is not linear.
I'd be grateful for any suggestion.