Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Algebraic and abelian solutions to the projective translation equation   Let $\mathbf{x}=(x,y)$. A projective 2-dimensional flow is a solution to a
2-dimensional projective translation equation (PrTE)
$(1-z)\phi(\mathbf{x})=\phi(\phi(\mathbf{x}z)(1-z)/z)$,
$\phi:\mathbb{C}^{2}\mapsto\mathbb{C}^{2}$. Previously we have found all
solutions of the PrTE which are rational functions. The rational flow gives
rise to a vector field $\varpi(x,y)\bullet\varrho(x,y)$ which is a pair of
2-homogenic rational functions. On the other hand, only very special pairs of
2-homogenic rational functions, as vector fields, give rise to rational flows.
The main ingredient in the proof of the classifying theorem is a reduction
algorithm for a pair of 2-homogenic rational functions. This reduction method
in fact allows to derive more results.
  Namely, in this work we find all projective flows with rational vector fields
whose orbits are algebraic curves. We call these flows abelian projective
flows, since either these flows are parametrized by abelian functions and with
the help of 1-homogenic birational plane transformations (1-BIR) the orbits of
these flows can be transformed into algebraic curves
$x^{A}(x-y)^{B}y^{C}\equiv\mathrm{const.}$ (abelian flows of type I), or there
exists a 1-BIR which transforms the orbits into the lines
$y\equiv\mathrm{const.}$ (abelian flows of type II), and generally the latter
flows are described in terms of non-arithmetic functions.
  Our second result classifies all abelian flows which are given by two
variable algebraic functions. We call these flows algebraic projective flows,
and these are abelian flows of type I. We also provide many examples of
algebraic, abelian and non-abelian flows.

34
The categories of this paper are: chem-ph
13
The categories of this paper are: quant-ph
11
The categories of this paper are: hep-lat
26
The categories of this paper are: bayes-an
5
The categories of this paper are: gr-qc
15
The categories of this paper are: nucl-ex
8
The categories of this paper are: hep-ex
17
The categories of this paper are: q-fin
4
The categories of this paper are: cond-mat
10
The categories of this paper are: q-bio
16
The categories of this paper are: stat
33
The categories of this paper are: plasm-ph
9
The categories of this paper are: nlin
18
The categories of this paper are: econ
27
The categories of this paper are: comp-gas
25
The categories of this paper are: patt-sol