Necessity conjecture for the CYBE solutions in the constructed ansatz

Let $\Psi^i(u)$ and $\Theta^i(u)$, $i=1,\ldots,n$, be the formal Laurent series or smooth functions occurring in the theorem, and let $\alpha,\beta\in{\mathbb R}$. Define

$$\varphi^{ij}(u,v)=\Theta^i(u)\Psi^j(v)-\Theta^j(v)\Psi^i(u).$$

The theorem gives a sufficient condition for $\varphi^{ij}(u,v)$ to solve the classical Yang–Baxter equation, namely

$$\Psi^s(u)\frac{\partial\Theta^i(u)}{\partial u^s}-\Theta^s(u)\frac{\partial\Psi^i(u)}{\partial u^s}=\alpha\Theta^i(u)+\beta\Psi^i(u),\quad i=1,\ldots,n.$$

Necessity conjecture. All solutions of the classical Yang–Baxter equation are described by the preceding ansatz and differential-equation condition. The statement is presented as the necessity part of the preceding sufficient-condition theorem; the source supplies no evidence resolving it.