Correct spelling, explanation: underlying is a present participle form of the word underlie. As you can see, it consists of two parts: the prefix under- and the verb lie. If you want to create a present participle of it, you have to change the second element to -ly and then add -ing. Underlaying on the other hand comes from the verb underlay. It’s a completely different word.

Definition of underlying:
1. verb (present participle), to be a cause of or strong influence on something
These ideas are underlying much of his work.
A theme of forbidden love was underlying much of her writing.
2. adjective, existing under the surface of something else
The underlying theme of the movie is very disturbing.
An underlying pessimism infuses all his novels.

Collocations with underlying:
Some most commonly used collocations include: the underlying motive and underlying reason.
This establishment’s underlying motive is to control others’ way of thinking.
The underlying reason for this event is to raise money for sick children.

Incorrect spelling, explanation: the correct version, underlying, can be easily confused with another word: underlaying. It’s important to remember that the second part of the first word – underlying – is a derivative of the verb lie, not lay. These two verbs are very similar, and that’s why many English users mistake them. You should be careful while spelling.