A simple design of a function f

Question

I am having trouble designing a function $f_i(x_1,x_2,x_3)$ for $i\in\{1,2,3\}$ such that

$ x_1\left(\frac{df_3}{dx_3} - \frac{df_1}{dx_1}\right) + x_2\left(\frac{df_1}{dx_1} - \frac{df_2}{dx_2}\right) + x_3\left(\frac{df_2}{dx_2} - \frac{df_3}{dx_3}\right) = \begin{cases} 0 & x_1=x_2=x_3\\>0 & \text{otherwise}\end{cases}$

for $(x_1,x_2,x_3)\in\mathbb{R}^3$.

Can anyone suggest a viable solution?

Edit: I also need all the $f_i$s to be the same functions of $x_i$s. For example, when $f_i=\sin(x_i+x_j)$ for $j={i+1}$ modulo $n$, then $f_1=\sin(x_1+x_2)$, $f_2=\sin(x_2+x_3)$ and $f_3=\sin(x_3+x_1)$.

Answer 1

With the given symmetry constraint,

$$f_1(x_1,x_2,x_3)=f_2(x_2,x_3,x_1)=f_3(x_2,x_1,x_2).$$

Then

$$\frac{\partial f_1}{\partial x_1}=\frac{\partial f_2}{\partial x_2}=\frac{\partial f_3}{\partial x_3}$$ and there is no solution.