P-symmetries of two-dimensional p-f vector fields (Q1198698)

Consider the initial value problem \(\dot y=V(y)\), \(y(0)=x\in\mathbb{F}^ n\) where \(V\) is a continuously differentiable vector field on \(\mathbb{F}^ n\) (\(\mathbb{F}\) is \(\mathbb{R}\) or \(\mathbb{C}\)). The corresponding flow \(\varphi\) is said to be a polynomial flow and \(V\) is said to be a \(p-f\) vector field if for each \(t\in\mathbb{R}\) the advance map \(\varphi^ t\) is polynomial. A polynomial map \(P: \mathbb{F}^ n\to\mathbb{F}^ n\) is called a polymorphism if it has a polynomial inverse. A diffeomorphism \(F:\mathbb{F}^ n\to\mathbb{F}^ n\) is called a symmetry of the field \(V\) if \(F'(x)V(x)=V(F(x))\) for all \(x\) in \(\mathbb{F}^ n\). A polymorphism which is a symmetry of \(V\) is called a \(p\)-symmetry of \(V\). The author shows that the group of \(p\)-symmetries of a two-dimensional \(p-f\) vector field can be explicitly found. It turns out that there exist only six basic \(p-f\) vector fields, so called Bass- Meisters normal forms of two-dimensional \(p-f\) vector fields to which every two-dimensional \(p-f\) vector field can be transformed by a polynomial transformation of coordinates.