Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On equivariant binary differential equations   This paper introduces the study of occurrence of symmetries in binary
differential equations (BDEs). These are implicit differential equations given
by the zeros of a quadratic 1-form, $a(x,y)dy^2 + b(x,y)dxdy + c(x,y)dx^2 = 0,$
for $a, b, c$ smooth real functions defined on an open set of $\mathbb{R}^2$.
Generically, solutions of a BDE are given as leaves of a pair of foliations,
and the appropriate way to define the action of a symmetry must depend not only
whether it preserves or inverts the plane orientation, but also whether it
preserves or interchanges the foliations. The first main result reveals this
dependence, which is given algebraically by a formula relating three group
homomorphisms defined on the symmetry group of the BDE. The second main result
adapts algebraic methods from invariant theory for representations of compact
Lie groups on the space of quadratic forms on $\mathbb{R}^n$, $n \geq 2$. With
that we obtain an algorithm to compute general expressions of quadratic forms.
Now, symmetric quadratic 1-forms are in one-to-one corrspondence with
equivariant quadratic forms on the plane, so these are treated here as a
particular case. We then apply the result to obtain the general forms of
equivariant quadratic 1-forms under each compact subgroup of the orthogonal
group $\mathbf{O}(2)$.

25
The categories of this paper are: patt-sol
5
The categories of this paper are: gr-qc
17
The categories of this paper are: q-fin
19
The categories of this paper are: eess
27
The categories of this paper are: comp-gas
34
The categories of this paper are: chem-ph
29
The categories of this paper are: funct-an
9
The categories of this paper are: nlin
33
The categories of this paper are: plasm-ph
32
The categories of this paper are: atom-ph
10
The categories of this paper are: q-bio
4
The categories of this paper are: cond-mat
14
The categories of this paper are: math-ph
24
The categories of this paper are: dg-ga
31
The categories of this paper are: ao-sci
28
The categories of this paper are: alg-geom