Annali di Matematica Pura ed Applicata

, Volume 163, Issue 1, pp 143–160 | Cite as

On the existence of global first integrals in the plane

  • Luisa Mazzi
  • Marco Sabatini


A plane, autonomous, noncritical differential system x′=f(x) of classCk is given. Under a suitable «discreteness» hypothesis on the behavior of orbits, the exeistence of a global first integral as regular as the field (provided k ≠ ω)is proved.


Differential System 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [ALGM]
    A. A.Andronov - E. A.Leontovich -I. I.Gordon - A. G.Maier,Qualitative Theory of Second Order Dynamic Systems, Israel Program for Scientific Translations (1971).Google Scholar
  2. [D]
    E. Digel, Über die Existenz von Integralen der partiellen Differentialgleichungf(x,y)(∂z/∂x)+g(x,y)(∂z/∂y)=0 i der Umgerbung eines singulären Punktes, Math. Zeit.,42 (1937), pp. 231–237.Google Scholar
  3. [F]
    R. Finn,On the uniformization of plane direction fields, and of second-order partial differential operators, Proc. Camb. Phil. Soc.,73 (1973), pp. 87–109.Google Scholar
  4. [K]
    E. Kamke, Über die partielle Differentialgleichungf(x,y)∂z/∂x+g(x,y)∂z/∂y=h(x,y), II, Math. Z.,42 (1936), pp. 287–300.Google Scholar
  5. [Ka]
    W. Kaplan,Regular curve-families filling the plane, I, Duke Math. J.,7 (1940), pp. 154–185.Google Scholar
  6. [M.1]
    L. Markus,Global structure of ordinary differential equations in the plane, Trans. Amer. Math. Soc.,76 (1954), pp. 127–148.Google Scholar
  7. [M.2]
    L. Markus,Topological types of polynomial differential equations, Trans. Amer. Math. Soc.,171 (1972), pp. 157–178.Google Scholar
  8. [MS]
    L. Mazzi -M. Sabatini,A characterization of centres via first integrals, Journ. Diff. Eq.,76 (1988), pp. 222–237.Google Scholar
  9. [Mu.1]
    M. P. Muller,An analytic foliation of the plane without weak first integrals of class C 1 Bol. Soc. Mat. Mex.,21 (1972), pp. 1–5.Google Scholar
  10. [Mu.2]
    M. P. Muller,Quelques proprietes des feuilletages polinomiaux du plan, Bol. Soc. Mat. Mex.,21 (1972), pp. 6–14.Google Scholar
  11. [NS]
    V. V.Nemytskii - V. V.Stepanov,Qualitative Theory of Differential Equations, Pure and Applied Mathematics XXI, Wiley Interscience (1969).Google Scholar
  12. SC]
    G.Sansone - R.Conti,Equazioni differenziali non lineari, Cremonese (1956).Google Scholar
  13. [Wa]
    T. Wazewski, Sur une problème de caractère intégral relatif à l'équation∂z/∂x++Q(x,y)∂z/∂y=0, Mathematica Cluj,8 (1934), pp. 103–116.Google Scholar
  14. [Wh]
    H. Whitney,Regular families of curves, Ann. Math.,34, 2 (1933), pp. 244–270.Google Scholar

Copyright information

© Fondazione Annali di Matematica Pura ed Applicata 1993

Authors and Affiliations

  • Luisa Mazzi
    • 1
  • Marco Sabatini
    • 2
  1. 1.Dipartimento di MatematicaPolitecnico di TorinoTorino
  2. 2.Dipartimento di MatematicaUniversità di TrentoPovo

Personalised recommendations