Now for the 'exactly two distinct' case:
Suppose n/M = 1/ck + 1/dk, where c and d are coprime and aren't both 1
<=> cdkn = M(c+d)
Now cd, c+d are coprime; n, M are coprime
<=> cd|M, (c+d)|kn, n|(c+d), M|cdk
<=> cdp = M, nq = (c+d), k = pq
So a decomposition is possible iff we can find coprime factors of M whose sum is a multiple of n (then we can choose k,p,q to fit).
I can't think of a general answer right now, but two things are immediate:
n = 1: decomposition is always possible
M prime: decompositions, if they exist, are unique
Come to think of it,
n = 1: since we can choose any pair of factors, if M is composite then the decomposition is not unique.
n = 2: decomposition is possible for M > 2, since M must have an odd factor f greater than 1, and f+1 is even. The decomposition is unique iff these are the only two odd factors of M => M is a power of 2 times a prime => M is prime.