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 $u=A\_{0}\left(x-x\_{0}\right)^{-1}$, where $A\_{0}=3\left(1+x\_{0}^{2}\right)t^{-1}$. I am looking for a [Lax pair](https://en.wikipedia.org/wiki/Lax_pair) $[\mathcal{L},\mathcal{M}]$ which satisfies the equation
$$\dot{\mathcal{L}}+[\mathcal{L},\mathcal{M}]=0,$$


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/comment(s) about the problem and/or the integrability of the PDE is appreciated. Also, are there any other (stronger) methods using which I can explicitly try and probe for the integrability of the PDE although it passed the Painleve test? 



Lax pair of an integrable non-linear PDE
You could try using the Wahlquist-Estabrook prolongation structure technique, per H.D. Wahlquist and F.B. Estabrook, J. Math. Phys **16** (1975) 1-7 (covering the Korteweg-deVries equation), & F.B. Estabrook and H.D. Wahlquist J. Math. Phys. **17** (1976) 1293-1297 (covering the nonlinear Schrödinger equation).


A non-trivial prolongation structure would be a signal that your equation is indeed integrable. To construct a Lax pair, you need to find an explicit representation of the prolongation structure: the paper by R. Dodd and A. Fordy Proc Roy. Soc. Lond. A385 (1983) 389-429 provides a method of doing this.


Since your PDE includes both independent variables in the coefficients, you may also wish to review the note by B.A. Kupershmidt, *Nonautonomous form of the theory of Lax equations*, Lettere al Nuovo Cimento **33** (1982) 103–107.


