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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 $[\mathcal{L},\mathcal{M}]$ 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.

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 $[\mathcal{L},\mathcal{M}]$ 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?

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 $[\mathcal{L},\mathcal{M}]$ 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/comment(s) about the problem and/or the integrability of the PDE is appreciated.

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 $[\mathcal{L},\mathcal{M}]$ 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.

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 $[\mathcal{L},\mathcal{M}]$ 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.

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 $[\mathcal{L},\mathcal{M}]$ 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/comment(s) about the problem and/or the integrability of the PDE is appreciated.