I need [an] expression that can help me [find] all [solutions].
Why are you interested in all solutions, Kodafafabe?
You have but one equation relating four unknowns. That's not enough information for any algebraic method.
You need an algorithm, to generate solutions. Perhaps later, after analyzing results, numerical patterns might be found for generating some solution
subsets. Otherwise, a computer program (using nested loops) is easiest. It will list all solutions within thresholds that you choose.
Also, you may have already realized that, once you obtain a solution, the Commutative Property tells us that permutations are also solutions. For example, if you were to discover four, distinct x,y,z,t values (eg: 6,11,22,33), then all 24 permutations are solutions.
Earlier, I'd realized that 1/3 = 4/12, so one x,y,z,t solution is 1/12, 1/12, 1/12, 1/12. A computer program confirms that no other solutions exist, unless one or more variables become larger than 12. For example, if we allow values up to 15, then there are nineteen solutions:
10, 10, 15, 15
10, 12, 12, 15
10, 12, 15, 12
10, 15, 10, 15
10, 15, 12, 12
10, 15, 15, 10
12, 10, 12, 15
12, 10, 15, 12
12, 12, 10, 15
12, 12, 12, 12
12, 12, 15, 10
12, 15, 10, 12
12, 15, 12, 10
15, 10, 10, 15
15, 10, 12, 12
15, 10, 15, 10
15, 12, 10, 12
15, 12, 12, 10
15, 15, 10, 10
Setting the upper threshold at 100 results in more than 2,200 solutions.
I'm not sure what else to suggest, without knowing your end goal.