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Sur les sous-groupes planaires des groupes des dispersions des équations différentielles linéaires du deuxième ordre. (French) Zbl 0554.34026
A group $${\mathfrak S}$$ consisting of real continuous functions of one real variable on the interval $$j=(-\infty,\infty)$$ is called planar if through each point of the plane $$j\times j$$ there passes just one element $$s\in {\mathfrak S}$$. Every differential oscillatory equation (Q): $$y''=Q(t)y$$ $$(t\in j=(-\infty,\infty)$$, $$Q\in C^{(0)})$$ admits functions, called the dispersions of (Q), that transform (Q) into itself. These dispersions are integrals of Kummer’s equation (QQ): $$-\{X,t\}+Q(X)X'{}^ 2(t)=Q(t)$$ and form a three-parameter group $${\mathfrak B}_ Q$$, known as the dispersion group of (Q). The increasing dispersions of (Q) form a three-parameter group $${\mathfrak B}^+_ Q(\subset {\mathfrak B}_ Q)$$ invariant in $${\mathfrak B}_ Q$$. The centre of the group $${\mathfrak B}^+_ Q$$ is an infinite cyclic group $${\mathfrak C}_ Q$$, whose elements, the central dispersions of (Q), describe the position of conjugate points of (Q).
The present paper contains new results concerning the algebraic structure of the group $${\mathfrak B}^+_ Q$$. It provides information on the following: (1) the existence and properties of planar subgroups of a given group $${\mathfrak B}^+_ Q$$ and (2) the existence and properties of the groups $${\mathfrak B}^+_ Q$$ containing a given planar group $${\mathfrak S}$$. The results obtained are: the planar subgroups of a given group $${\mathfrak B}^+_ Q$$ form a system depending on two constants, $${\mathcal S}Q$$, such that $$\cap {\mathfrak S}={\mathfrak C}_ Q$$ for all $${\mathfrak S}\in {\mathcal S}Q$$. The equations (Q) whose groups $${\mathfrak B}^+_ Q$$ contain the given planar group $${\mathfrak S}$$ form a system dependent on one constant, $${\mathcal Q}S$$, such that $$\cap {\mathfrak B}^+_ Q={\mathfrak S}=\cup {\mathfrak C}_ Q$$ for all (Q)$$\in {\mathcal Q}S$$.

##### MSC:
 34C20 Transformation and reduction of ordinary differential equations and systems, normal forms 34A30 Linear ordinary differential equations and systems
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##### References:
 [1] Kuczma, Functional Equations in a Single Variable (1968) · Zbl 0196.16403 [2] Blanton, Arch. Math. (Brno) 18 pp 121– (1982) [3] Boruvka, Linear Differential Transformations of the Second Order (1971) [4] Boruvka, Ann. Polon. Math. 42 pp 27– (1982)
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