The hyperelliptic curve cryptography is based on the arithmetic in the Jacobian of a curve. In classical mechanics well-known cryptographic algorithms and protocols can be very useful for construct auto-Bäcklund transformations, discretization of continuous flows and study of integrable systems with higher order integrals of motion. We consider application of a standard arithmetic of divisors on genus two hyperelliptic curve for the construction of new auto-Bäcklund transformations for the Hénon-Heiles system. Another type of auto-Bäcklund transformations associated with equivalence relations between unreduced divisors and the construction of the new integrable systems in the framework of the Jacobi method are also discussed.