1, 1 and 7. The sum is 3.75 – take it or leave it.

This person has found an actual solution:

https://mlzeng.com/an-interesting-equation.html

x = 36875131794129999827197811565225474825492979968971970996283137471637224634055579

y = 154476802108746166441951315019919837485664325669565431700026634898253202035277999

z = 4373612677928697257861252602371390152816537558161613618621437993378423467772036

This is apparently the smallest solution.

The derivation of the solution itself requires an understanding of Diophantine equations and elliptical curves, which I will not pretend to understand well enough to summarize.