Monogamous latin squares

We show for all n∉{1,2,4} that there exists a latin square of order n that contains two entries γ1 and γ2 such that there are some transversals through γ1 but they all include γ2 as well. We use this result to show that if n>6 and n is not of the form 2p for a prime p⩾11 then there exists a latin square of order n that possesses an orthogonal mate but is not in any triple of MOLS. Such examples provide pairs of 2-maxMOLS.