New algebraic theories?

I. \(\large x+y = \sqrt{z^2+2xy}\)

II. \(\large x+y = \sqrt[3]{z^3+3xy(x+y)}\)

III. \(\large \left(\dfrac{x}{z}\right)^2+\left(\dfrac{y}{z}\right)^2=1\)

IV. \(\large \left(\dfrac{x}{z}\right)^4+\left(\dfrac{y}{z}\right)^4=1\)

V. \(\large \left(\dfrac{x}{z}\right)^4+\left(\dfrac{y}{z}\right)^4 = \dfrac{z^4-2x^{2}y^{2}}{z^4} \)

Which equation(s) above has/have infinitely many positive integer solutions for \(x\) , \(y\) and \(z\)?

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