Monogamous latin squares

We show for all n ∉ { 1 , 2 , 4 } that there exists a latin square of order n that contains two entries γ 1 and γ 2 such that there are some transversals through γ 1 but they all include γ 2 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.