We present new orbital elements for two symbiotic systems involving a giant of spectral type S, namely V420 Hya and ER, Del. These orbital elements are compared with existing elements for S-type binaries and for symbiotic binaries with M-giant primaries. It is shown that among the S-type binaries, most of the short-period systems (with P in the range 300 - 1000 d) exhibit some kind of symbiotic activity (H alpha emission, UV continuum, hard X-rays), but symbiotic systems are not restricted to the short-period systems. The complex and varying H alpha profile of V420 Hya has been decomposed to several components (broad emission with sigma similar to 140 km/s, narrow emission with sigma similar to 60 km/s, and narrow absorption components). Their orbital modulation reveals that the broad emission is located close to the companion (under the hypothesis of a system with a mass ratio M-g/M-c = 2), and that this broad emission is mutilated by absorption from matter located along the line of sight and flowing towards the observer (faster than the giant) at all orbital phases.