  We investigate the influence of a self-propelling, out-of-equilibrium active
particle on generalized elastic systems, including flexible and semiflexible
polymers, fluid membranes, and fluctuating interfaces, while accounting for
long-ranged hydrodynamic effects. We derive the fractional Langevin equation
governing the dynamics of the active particle, as well as that of any other
passive particle (or probe) bound to the elastic system. This equation
demonstrates analytically how the active particle dynamics is influenced by the
interplay of both the non-equilibrium force and of the viscoelastic
environment. Our study explores the diffusional behavior emerging for both the
active particle and a distant probe.The active particle undergoes three
different surprising and counterintuitive regimes identified by the distinct
dynamical time-scales: a pseudo-ballistic initial phase, a drastic decrease of
the mobility and an asymptotic subdiffusive regime.
