Event Overlap Analyzer

You are given an array of events where each event is represented by a pair of integers [start, end]. The start time is always less than the end time. The times are in arbitrary units (e.g., seconds, minutes), and the events may overlap.

Write a function that takes this array as input and returns two things:

1. The maximum number of events that overlap at any point in time.
2. A list of one or more continuous time intervals during which the number of overlapping events is exactly the maximum found.

For example, if the maximum overlap occurs from time 5 to 8 and again from 10 to 12, your function should return these intervals along with the maximum overlap count.

Make sure your solution handles edge cases such as events with the same start or end times, and that it works efficiently even for a large number of events.

Your solution should be self-contained and include appropriate tests to demonstrate its correctness.