  We propose a system of equations that defines Weierstrass--Jacobi's eta- and
theta-constant series in a differentially closed way. This system is shown to
have a direct relationship to a little-known dynamical system obtained by
Jacobi. The classically known differential equations by Darboux--Halphen,
Chazy, and Ramanujan are the differential consequences or reductions of these
systems. The proposed system is shown to admit the Lagrangian, Hamiltonian, and
Nambu formulations. We explicitly construct a pencil of nonlinear Poisson
brackets and complete set of involutive conserved quantities. As byproducts of
the theory, we exemplify conserved quantities for the Ramamani dynamical system
and quadratic system of Halphen--Brioschi.
