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> How to solve the system of partial differential equations $$u\_{1x\_1}=u\_{2x\_2}=u\_{3x\_3=0}$$$$\frac{\partial u\_i}{\partial x\_j}+\frac{\partial u\_j}{\partial x\_i}=0$$ for $i=1,2,3$
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I tried so hard and it becomes so complicated. 
I first conclude $u\_1=u\_1(x\_2,x\_3), u\_2=u\_2(x\_1,x\_3),u\_3=u\_3(x\_1,x\_2)$.
Then by $\frac{\partial u\_1}{\partial x\_2}+\frac{\partial u\_2}{\partial x\_1}=0$, I get $u\_1(x\_2,x\_3)=c\_1(x\_3)x\_2+b\_1(x\_3)$, and $u\_2(x\_1,x\_3)=-c\_1(x\_3)x\_1+b\_2(x\_3)$. I did these for three of the system of equations and in the end it becomes very complicated. I cannot solve for the coefficient $c\_1,b\_1,c\_2,b\_2$.etc. at all. 

Could someone kindely provide some help ? Thanks!