Planar vector fields with a given set of orbits (Q657760)

The authors first present a general result on determining all \(C^1\) planar vector fields with a given set of orbits of the form \(y-y_i(x)=0\) (\(i=1,\dots,s\)), where \( y_i(x)\) are \(C^1\) functions satisfying some assumptions. Then they study in detail few cases when the orbits are branches of an algebraic curve. In particular, they consider the quadratic vector field associated to the system \[ \dot x=p_0 y^2+p_1 y+p_2, \quad \dot y=q_0 y^2+q_1 y+q_2, \tag{1} \] where \(p_j=\sum_{n=0}^j p_{jn} x^n, \;q_j=\sum_{n=0}^j q_{jn} x^n, \;j=1, 2,\) for which \[ g=\sum_{j=0} ^S a_j (x) y^{S-j}=0 \tag{2} \] is the unique irreducible algebraic invariant curve. It is proven that if the curve \(g=0\) has \(S>1\) branches with respect to the variable \(y\), then deg\,\(g\leq 4S\). The authors also present the following two conjectures: - if system (1) admits a unique invariant irreducible algebraic curve (2), then deg\,\(g\leq 3 S\); - if system (1) with a unique invariant irreducible algebraic curve (2) does not admit a rational first integral, then deg\,\(g\leq 12\).