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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)\(\in {\mathcal Q}S\).

34C20 Transformation and reduction of ordinary differential equations and systems, normal forms
34A30 Linear ordinary differential equations and systems
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[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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