Lax pair of an integrable non-linear PDE

Question

The following is a fourth-order non-linear PDE that passes the Painleve integrability test

$$\left(1+x^{2}\right)^{2}u_{xxxx} + 8x\left(1+x^{2}\right)u_{xxx} + 4\left(1+3x^{2}\right)u_{xx}$$

$$+ t\left(2xuu_{xx}+ \left(1+x^{2}\right)\left(uu_{xx}\right)_{x} - 4\left(1+3x^{2}\right)u_{xxt} - 4x\left(1+x^{2}\right)u_{xxxt}\right)=0,$$

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.