This paper is devoted to a study of special subgroups of automorphism groups of Kronrod-Reeb graphs of Morse functions on 2-torus T 2 which arise from actions of diffeomorphisms preserving a given Morse function on T 2. In this paper we give a full description of such classes of groups. S(f, X) = {h ∈ D(M, X) | f • h = f }, O(f, X) = {f • h | h ∈ D(M, X)} the stabilizer and the orbit of f ∈ C ∞ (M). Endow C ∞ (M) and D(M, X) with strong Whitney topologies; for a fixed f ∈ C ∞ (M) these topologies induce some topologies on S(f, X) and O(f, X). By D id (M, X), S id (f, X), and O f (f, X) we denote connected components of the identity map id M of D(M, X), S(f, X), and the component of O(f, X)

Let M be a connected orientable compact surface, f : M → R be a Morse function, and D id (M ) be the group of difeomorphisms of M isotopic to the identity. Denote by S (f ) = {f • h = f | h ∈ D id (M )} the subgroup of D id (M ) consisting of difeomorphisms "preserving" f , i.e. the stabilizer of f with respect to the right action of D id (M ) on the space C ∞

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Let f : S 2 → R be a Morse function on the 2-sphere and K be a connected component of some level set of f containing at least one saddle critical point. Then K is a 1-dimensional CW-complex cellularly embedded into S 2 , so the complement S 2 \ K is a union of open 2-disks D 1 , . . . , D k . Let S K (f ) be the group of isotopic to the identity diffeomorphisms of S 2 leaving invariant K and also each level set f −1 (c), c ∈ R. Then each h ∈ S K (f ) induces a certain permutation σ h of those disks. Denote by G = {σ h | h ∈ S K (f )} be the group of all such permutations. We prove that G is isomorphic to a finite subgroup of SO(3).

Let f : S 2 → R be a Morse function on the 2-sphere and K be a connected component of some level set of f containing at least one saddle critical point. Then K is a 1-dimensional CW-complex cellularly embedded into S 2 , so the complement S 2 \ K is a union of open 2-disks D 1 , . . . , D k . Let S K (f ) be the group of isotopic to the identity diffeomorphisms of S 2 leaving invariant K and also each level set f −1 (c), c ∈ R. Then each h ∈ S K (f ) induces a certain permutation σ h of those disks. Denote by G = {σ h | h ∈ S K (f )} the group of all such permutations. We prove that G is isomorphic to a finite subgroup of SO(3).