COCA shows enough results for 'equal-sized' that serve as an adjective. So, in your case, it could be...
There is a desert with two equal-sized water pools.
We then found cutoff points within each category to form 3 nearly equal-sized groups (COCA).
or, another way to say this is the last option you quoted. Use 'of equal size' after the noun in concern.
There is a desert with two water pools of equal size.
Break the dough into 15 portions of equal size (COCA).
The words 'equal-size' is also possible but less frequently shown.
Fetch the dough from the refrigerator and cut it into eight equal-size pieces (COCA)
Finally, 'equally-sized' is an improper use of those words. Don't use it.
'Equal size' without hyphen is also found. Again, less frequently used.
By cutting between the uprights, I can get four equal size pieces of firewood (COCA)
So, to sum up:
There is a desert with two equal size water pools -less frequent
There is a desert with two equal-size water pools -less frequent
There is a desert with two equally-sized water pools -improper
There is a desert with two water pools of equal size -proper (and preferred?). And adding...
There's a desert with two equal-sized water pools -proper (and frequent?).