6 problems

Non-integrability conjecture. The system is non-integrable for the case $2\xi=3(1-C)$; in particular, it does not admit an additional real meromorphic first integral functionally i…

Let $\lambda$ and $E$ be the parameters of the axisymmetric Bianchi IX cosmological model, and let $\gamma_1,\gamma_2\in\mathbb{C}$ be the constants appearing in the differential G…

Let $V_{n,1}$ denote the colinear $n$-body potential, and consider its Darboux points and the eigenvalues of its Hessian matrix at those points. Using a real algebraic geometry com…

Let $m$ be the parameter in the Friedmann–Robertson–Walker Hamiltonian, let $\Lambda$ and $\lambda$ be its parameters, and let $\mu_i(p)$, for $i=1,2,3$, denote the three parameter…

Let $K_{4,36}$ denote the order-five variational-equation entry obtained by transporting the jet along the cycle $\gamma=\gamma_-^{-1}\gamma_+^{-1}\gamma_-\gamma_+$. Let…

Let $n\in\mathbb{N}$ and consider the algebraic surface … Here, non-integrability means the absence of meromorphic first integrals for its geodesic flow. Non-integrability conjectu…