Integrability of Truncated Hořava–Lifshitz Mixmaster Model in Rosenhain Functions

The mixmaster Hořava–Lifshitz model belongs to generalized Euclidean Toda chains with 28 vectors. The longest three vectors of the spectrum play a dominant role in studying its dynamics. The truncated cosmological model is presented as a periodic three-particle Toda chain. It is associated with an affine Kac–Moody Lie algebra \(A_{2}^{+}\). According to the Adler–van Moerbeke criterion, the truncated Hamiltonian system is algebraically completely integrable. The phase curves wrap a torus of genus 2. The Jacobi problem of inversion of ultraelliptic integrals is solved by using theta-functions of two variables. The solutions of the dynamical problem are expressed in rational functions of Rosenhain theta-functions. They are four-periodic functions.