Integral solutions only!!

Algebra Level 5

What is the number of positive integral solutions of the equation \[\begin{vmatrix} { y }^{ 3 }+1 & { y }^{ 2 }z & { y }^{ 2 }x \\ y{ z }^{ 2 } & { z }^{ 3 }+1 & { z }^{ 2 }x \\ y{ x }^{ 2 } & { x }^{ 2 }z & { x }^{ 3 }+1 \end{vmatrix}=11\]

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