Anharmonic solutions to the Riccati equation and elliptic modular functions

We study the irreducible algebraic equationx^{n}+a_{1}x^{n-1}+\cdots+a_{n}=0,\quad\text{with ${n\geq 4}$,}on the differential field{(\mathbb{F}=\mathbb{C}(t),\delta=\frac{d}{dt})}. We assume that a root of the equation is a solution to the Riccati differential equation{u^{\prime}+B_{0}+B_{1}u+B_{2}u^{2}=0}, where{B_{0}},{B_{1}},{B_{2}}are in{\mathbb{F}}.We show how to construct a large class of polynomials as in the above algebraic equation, i.e., we prove that there exists a polynomial{F_{n}(x,y)\in\mathbb{C}(x)[y]}such that for almost{T\in\mathbb{F}\setminus\mathbb{C}}, the algebraic equation{F_{n}(x,T)=0}is of the same type as the above stated algebraic equation. In other words, all its roots are solutions to the same Riccati equation. On the other hand, we give an example of a degree 3 irreducible polynomial equation satisfied by certain weight 2 modular forms for the subgroup{\Gamma(2)}, whose roots satisfy a common Riccati equation on the differential field{(\mathbb{C}(E_{2},E_{4},E_{6}),\frac{d}{d\tau})}, with{E_{i}(\tau)}being the Eisenstein series of weighti. These solutions are related to a Darboux–Halphen system. Finally, we deal with the following problem: For which “potential”{q\in\mathbb{C}(\wp,\wp^{\prime})}does the Riccati equation{\frac{dY}{dz}+Y^{2}=q}admit algebraic solutions over the differential field{\mathbb{C}(\wp,\wp^{\prime})}, with{\wp}being the classical Weierstrass function? We study this problem via Darboux polynomials and invariant theory and show that the minimal polynomial{\Phi(x)}of an algebraic solutionumust have a vanishing fourth transvectant{\tau_{4}(\Phi)(x)}.

Bianchi-IX, Darboux–Halphen and Chazy–Ramanujan

Bianchi-IX four metrics are SU(2) invariant solutions of vacuum Einstein equation, for which the connection-wise self-dual case describes the Euler top, while the curvature-wise self-dual case yields the Ricci flat classical Darboux–Halphen system. It is possible to see such a solution exhibiting Ricci flow. The classical Darboux–Halphen system is a special case of the generalized one that arises from a reduction of the self-dual Yang–Mills equation and the solutions to the related homogeneous quadratic differential equations provide the desired metric. A few integrable and near-integrable dynamical systems related to the Darboux–Halphen system and occurring in the study of Bianchi-IX gravitational instanton have been listed as well. We explore in details whether self-duality implies integrability.