The following is a fourth-order non-linear PDE that passes the Painleve integrability test
$$\left(1+x^{2}\right)^{2}u_{xxxx} + 8\eta\left(1+x^{2}\right)u_{xxx} + 4\left(1+3x^{2}\right)u_{xx}=0,$$

where $u=u(x,t)$. The leading-order behaviour of this PDE is of the form $u\approx\left(x-x_{0}\right)^{-1}$. I am looking for a Lax pair $[\mathcal{L},\mathcal{M}]$ which satisfies the equation
$$\dot{\mathcal{L}}=[\mathcal{L},\mathcal{M}],$$

but not able to find the same. I suspect that this may be due to the fact that although few systems may pass the Painleve test, they need not be integrable. Any help is appreciated.