System of equation

[Agung]

Find all positive real triples (x,y,z) which satisfy:
2x^3 = 2y(x^2+1) - (z^2+1)
2y^4 = 3z(y^2+1) - 2(x^2+1)
2z^5 = 4x(z^2+1)- 3(y^2+1)

 

[mmm4444bot]

I can't solve this system, but substitution shows that the typical guess of (1,1,1) works.

I wanted to graph the system, too, but I can solve only the first two equations for z.