Solution

Functional dependencies:

$U \rightarrow V$

$VW \rightarrow X$

$Y \rightarrow W$

$X \rightarrow U$

Start with $UY$:

$U \rightarrow V$

$Y \rightarrow W$

Thus we get $V,W$

Then

$VW \rightarrow X$

Then

$X \rightarrow U$

So closure:

$UY^+ = {U,V,W,X,Y}$

Add $Z$ to cover all attributes.

Thus minimal key:

$UYZ$

Similarly:

$VYZ$ and $XYZ$

Thus candidate keys:

$UYZ,; VYZ,; XYZ$