  We investigate a sample of 152 dusty sources at 1:5 < z < 2:5 to understand
the connection of enhanced Star-Formation-Rate (SFR) and
Black-Hole-Accretion-Rate (BHAR). The sources are Herschel-selected, having
stellar masses M*>10^10Msun and SFR (100-1000Msun/yr) elevated(>4?) above the
star-forming "main sequence", classifying them as Starbursts (SB). Through a
multiwavelength fitting approach (including a dusty torus component), we
divided the sample into active SBs (dominated by an AGN emission, SBs-AGN, ?
23% of the sample) and purely star-forming SBs (SBs-SFR). We visually inspected
their HST/UV-restframe maps: SBs-SFR are generally irregular and composite
systems; ? 50% of SBs-AGN are instead dominated by regular compact
morphologies. We then found archival ALMA continuum counterparts for 33
galaxies (12 SBs-AGN and 21 SBs-SFR). For these sources we computed dust
masses, and, with standard assumptions, we also guessed total molecular
gas-masses. SBs turn to be gas rich systems (fgas = Mgas=Mgas/(Mgas+M*) \sim
20%-70%), and the gas fractions of the two SB classes are very similar (fgas =
43 +/-4% and fgas = 42+/- 2%). Our results show that SBs are consistent with a
mixture of: 1) highly star-forming merging systems (dominating the SBs-SFR),
and 2) primordial galaxies, rapidly growing their M* together with their Black
Hole (mainly the more compact SBs-AGN). Anyway, feedback effects have not
reduced their fgas yet. Indeed, SBs at z = 2, with relatively low bolometric
AGN luminosities in the range 10^44 < Lbol(AGN) < 10^46 erg/s (compared to
bright optical and X-ray quasars), are still relatively far from the epoch when
the AGN feedback will quench the SFR in the host and will substantially depress
the gas fractions.
