The system in question is: $x+y^b=y+z^c=z+x^a$ where $x,y,z$ are odd primes that are pairwise distinct and $a,b,c$ are positive integers.
From basic rearrangement I found that from $a,b,c$ only one can be even, this can be seen with:
$x+y^{2b} = y + z^{2c}$ $\rightarrow y^{2b} - z^{2c} = y - x \rightarrow (y^{b} - z^{c})(y^b + z^c) = y - x$
Which is not possible, there is a little more work that needs to be done to show this for all combinations though. This is where I got stuck.
Is such a system possible? And are there any methods to make progress on this problem?