Correct spelling


Correct spelling, explanation: heavily is the correct spelling as it is an adverb made from the adjective heavy, which ends with y. Therefore, according to the rule, we change this y to i and add -ly (adverbial ending) to receive heavily – an adverb. This means that heaviliy is incorrect: there is no point in adding another i in between.

Definition of heavily:
adverb, in an abundant manner; a lot
This project has to be heavily modified to be successful.
It was raining heavily yesterday.

Incorrect spelling


Incorrect spelling, explanation: heaviliy is incorrect as the correct adverbial suffix is -ly, not -liy. Heavy plus -ly is heavily, not heaviliy. Therefore heavily is the correct form and others are mistakes and probably just typos.