Recall the necessary definitions. We give them in the general setting, though for our needs everywhere below we may take M = S 2 . Here and below B ⊂ R k (base of a family) is a topological open ball.Denote by Vect(M ) the set of C 3 -smooth vector fields on M . Definition 1. A family of vector fields on a manifold M with the base B is a vector field V on B × M tangent to the fibers { α } × M , α ∈ B. The dimension of a family is the dimension of its base. An equivalent definition. Definition 2. A family of vector fields on M with the base B is a smooth map V : B → Vect(M ). The equivalence is obvious. Denote by V k (M ) the space of k-parameter families of vector fields on M which are C 3 smooth as vector fields on B × M . Definition 3. A family of vector fields is transversal to a Banach submanifold T of Vect(M ) provided that the corresponding map V : B → Vect(M ) is transversal to T. Definition 4. Two vector fields v andṽ on a manifold M are called orbitally topologically equivalent, if there exists a homeomorphism M → M that links the phase portraits of v andṽ, that is, sends orbits of v to orbits ofṽ and preserves their time orientation. Definition 5. Two families of vector fields { v α | α ∈ B }, {ṽα |α ∈B } on M are called weakly topologically equivalent if there exists a map

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