Let $$f,g: M(\phi _1)\rightarrow M(\phi _2)$$ be fibre-preserving maps over the circle, $$S^1$$, where $$M(\phi _1)$$ and $$M(\phi _2)$$ are fibre bundles over $$S^1$$ and the fibre is the torus, T. The main purpose of this work is to classify the pairs of maps (f, g) which can be deformed by fibrewise homotopy over $$S^1$$ to a coincidence-free pair $$(f^{\prime },g^{\prime })$$, $$f^{\prime },g^{\prime }: M(\phi _1)\rightarrow M(\phi _2)$$. In general, the classification of such pairs of maps is equivalent to finding solutions for an equation in the free group $$\pi _2(T,T-1)$$, called the main equation. In certain situations, it is appropriate to study the main equation in the abelianization of $$\pi _2(T, T-1)$$ or on some quotients of this group, since, if the equation in one of these quotients does not admit solution, then the original equation also does not admit solution. In this case, it is not possible to obtain the desired deformability.