You're right that it is the presence of 'can' that determines no 's', E, but there is no "perfect tense".
[And actually, the perfect isn't a tense, it's an aspect.
When a lexical verb - climb, run, jump, look, etc
is used with a modal verb - can, could, may, might, will, would, should, shall, etc
third person - he, she, it
never attracts an 's'.