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The projective translation equation and rational plane flows. II. Corrections and additions


In this second part of the work, we correct the flaw which was left in the proof of the main Theorem in the first part. This affects only a small part of the text in this first part and two consecutive papers. Yet, some additional arguments are needed to claim the validity of the classification results. With these new results, algebraic and rational flows can be much more easily and transparently classified. It also turns out that the notion of an algebraic projective flow is a very natural one. For example, we give an inductive (on dimension) method to build algebraic projective flows with rational vector fields, and ask whether these account for all such flows. Further, we expand on results concerning rational flows in dimension 2. Previously we found all such flows symmetric with respect to a linear involution \(i_{0}(x,y)=(y,x)\). Here we find all rational flows symmetric with respect to a non-linear 1-homogeneous involution \(i(x,y)=(\frac{y^2}{x},y)\). We also find all solenoidal rational flows. Up to linear conjugation, there appears to be exactly two non-trivial examples.

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Correspondence to Giedrius Alkauskas.

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The research of the author was supported by the Research Council of Lithuania Grant No. MIP-072/2015.

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Alkauskas, G. The projective translation equation and rational plane flows. II. Corrections and additions. Aequat. Math. 91, 871–907 (2017).

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Mathematics Subject Classification

  • Primary 39B12
  • 14E07
  • Secondary 35F05
  • 37E35


  • Translation equation
  • Flow
  • Projective geometry
  • Rational functions
  • Rational vector fields
  • Iterable functions
  • Birational transformations
  • Involutions
  • Cremona group
  • Linear ODE
  • Linear PDE