Longest substring with consecutive repetitions
From what I understand you are looking for maximal substrings of the form $y^k$ for $k\ge 2$. By some this is called a repetition in a string. One of the first to give an algorithm for finding these is Maxime Crochemore: An optimal algorithm for computing all the repetitions in a word, Inform. Process. Lett. 12 (1981) 244-248. From what I understand his approach has been reformulated in terms of suffix trees and arrays.