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Let Φ = (f, g):(Cn+ 1,0) → (C2, 0) be a pair of holomorphic germs with no blowing up in codimension 0. (Two examples are the following: Φ defines an isolated complete intersection singularity; g = lN where ℓ is a generic linear form with respect to f and N> 0.) We study how the Milnor fibrations of the germs ϕ(α:β) = α g-β f are related to each other when (α:β) varies in P1. More precisely, we construct isotopic subfibrations or subfibres of the Milnor fibrations of any two such germs. The proofs are based on the precise study of the subdiscs of complex lines meeting a fixed complex plane curve germ transversally, generalizing Lê's work on the Cerf diagram. 2000 Mathematical Subject Classification: 32S55, 32S15, 32S30