One part that repeats a single pitch for the entire tune, and another part repeats a pitch a major or minor third away.
One part oscillates between the root and third of a major or minor chord, and another part oscillates between the third and fifth.
I think the "real" question is whether a musically "interesting" counterpoint can be written. And in this case, I believe the answer is no.
In this particular case, we're talking about a canon: a single pattern that plays against itself. So we need to construct a musically interesting melody that can be phase shifted against itself and remain a classically rule-adherent canon.
The melody, in this case, will be required to include seconds — a classical melody entirely of skips would not be aesthetically acceptable. Further, to avoid just oscillating between the same two pitches (which wouldn't meet the "rules"), there will have to be a passage with at least two consecutive seconds, either both ascending or both descending. Let's make them both ascending, since the descending case will lead to the same problem.
In classical counterpoint, when two voices form a second, the should either resolve to a third or a unison. However, there will be phases in which the seconds between the two voices do not resolve properly: for example when the first and second notes in one voice overlap with the second and third notes in the other voice.
There is a related topic that might be of interest: tiling rhythmic canons. These are rhythmic patterns which, when presented in canon with themselves, fill the entire metric space. For more see: What is rhythmic counterpoint? and What's "species counterpoint"? Are there any other types of counterpoint?.