## Abstract

Let **x** = (*x*, *y*). A *plane flow* is a function \({F(\mathbf{x}, t) : \mathbb{R}^2 \times \mathbb{R} \mapsto \mathbb{R}^{2}}\) such that *F*(*F*(**x**, *s*), *t*) = *F*(**x**, *s* + *t*) for (almost) all real numbers *x*, *y*, *s*, *t* (the function *F* might not be well-defined for certain **x**, *t*). In this paper we investigate rational plane flows which are of the form \({F(\mathbf{x}, t) = \phi(\mathbf{x}t)/t;}\) here \({\phi}\) is a pair of rational functions in 2 real variables. These may be called *projective flows*, and for a description of such flows only the knowledge of Cremona group in dimension 1 is needed. Thus, the aim of this work is to completely describe over \({\mathbb{R}}\) all rational solutions of the two dimensional translation equation \({(1 - z)\phi(\mathbf{x}) = \phi(\phi(\mathbf{x}z)(1 - z)/z)}\). We show that, up to conjugation with a 1-homogenic birational plane transformation (1-BIR), all solutions are as follows: a zero flow, two singular flows, an identity flow, and one non-singular flow for each non-negative integer *N*, called *the level* of the flow. The case *N* = 0 stands apart, while the case *N* = 1 has special features as well. Conjugation of these canonical solutions with 1-BIR produce a variety of flows with different properties and invariants, depending on the level and on the conjugation itself. We explore many more features of these flows; for example, there are 1, 4, and 2 essentially different symmetric flows in cases *N* = 0, *N* = 1, and *N* ≥ 2, respectively. Many more questions related to rational flows will be treated in the second part of this work.

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The author gratefully acknowledges support by the Lithuanian Science Council whose postdoctoral fellowship is being funded by European Union Structural Funds project “Postdoctoral Fellowship Implementation in Lithuania”.

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Alkauskas, G. The projective translation equation and rational plane flows. I.
*Aequat. Math.* **85**, 273–328 (2013). https://doi.org/10.1007/s00010-013-0190-1

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DOI: https://doi.org/10.1007/s00010-013-0190-1

### Mathematics Subject Classification (2000)

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

### Keywords

- Translation equation
- flow
- projective geometry
- rational functions
- rational vector fields
- iterable functions
- birational transformations
- involutions
- Cremona group
- linear ODE
- linear PDE