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Solving a Killing vector problem, in General Relativity, I got the following PDE system:

[tex]

\frac{\partial X^0}{\partial x}=0

[/tex]

[tex]

\frac{\partial X^1}{\partial y}=0

[/tex]

[tex]

\frac{\partial X^2}{\partial z}=0

[/tex]

[tex]

\frac{\partial X^0}{\partial y} + \frac{\partial X^1}{\partial x}=0

[/tex]

[tex]

\frac{\partial X^0}{\partial z} + \frac{\partial X^2}{\partial x}=0

[/tex]

[tex]

\frac{\partial X^1}{\partial z} + \frac{\partial X^2}{\partial y}=0

[/tex]

where [itex] X^a , a=0,1,2[/itex] are the three components of the Killing vector that I am looking for. I have spend a lot of time trying to solve this system but I am not getting any solution.

Thanks for any idea.

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# Help with a PDE system

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