Let <$Ef~:~M~symbol О~roman bold R> be a Morse function on a smooth closed surface, V be a connected component of some critical level of f, and <$Eepsilon sub V> be its atom. Let also S(f) be a stabilizer or the function f under the right action of the group of dilfeomorphisms Diff(M) on the space of smooth functions on M, and <$ES sub V (f)~=~left { h~symbol <174>~S(f)|h(V)~=~V right }>. The group SV(f) acts on the set <$Epi sub 0 del epsilon sub V> of connected components of the boundary of <$Eepsilon sub V>. Therefore we have a homomorphism <$Esymbol f~:~S(f)~symbol О~roman Aut ( pi sub 0 del epsilon sub V )>. Let also <$EG~=~symbol f S(f)> be the image of S(f) in <$Eroman Aut ( pi sub 0 del epsilon sub V)>. Suppose that the inclusion <$Edel epsilon sub V ~symbol <172>~M\V> induces a bijection <$Epi sub 0 del epsilon sub V ~symbol О~pi sub 0 (M\V)>. Let H be a subgroup of G. We present a sufficient condition for existence of a section <$Es~:~H~symbol О~S sub V (f)> of the homomorphism <$Esymbol f>, so, the action of H on <$Edel epsilon sub V> lifts to the H-action on M by f-preserving diffeomorphisms of M. This result holds for a larger class of smooth functions <$Ef~:~M~symbol О~roman bold R> having the following property: for each critical point z of f the germ of f at z is smoothly equivalent to a homogeneous polynomial <$E{ roman bold R} sup 2 ~symbol О~roman bold R> without multiple linear factors.

• Фещенко Богдан Григорович (фізико-математичні науки)

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