LetKbe a non-dyadic local field andAits ring of integers. First we notice that the classification of unimodularA-bilinear forms and of systems of two unimodular symmetricA-bilinear forms are both reduced to the classification of hermitian forms over the category of double isomorphisms over projectiveA-modules ; the duality is nevertheless different for each case. Then we use the methods of reduction and transfer to construct some examples of non isometric unimodular forms (or system of unimodular symmetric forms) having the same asymmetry and becoming isometric overK.