Existence of immovability lines of a partial mapping of Euclidean space E5

It is considered a set of smooth lines such that through a point X ∈ Ω passed one line of given set in domain Ω ⊂ E 5 . The moving frame ℜ = ( X , e → i , ) ( i , j , k = 1 , 5 ¯ ) is frame of Frenet for the line ω 1 of the given set. Integral lines of the vector fields e → i are formed net ∑ 5 of Frenet. There exists a point F 5 4 ∈ ( X , e → 5 ) on the tangent of the line ∑ 5 . When a point X is shifted in the domain Ω the point F 5 4 describes it’s domain Ω 5 4 in E 5 . It is defined the partial mapping f 5 4 : Ω → Ω 5 4 , such that f 5 4 : ( X ) = F 5 4 . Necessary and sufficient conditions of immovability and degeneration of lines ( X , e → 1 ) , ( X , e → 2 ) and ( X , e → 3 ) in partial mapping f 5 4 are obtained.