We propose a mechanism for the generation of pure superconducting
spin-current carried by equal-spin triplet Cooper pairs in a superconductor (S)
sandwiched between a ferromagnet (F) and a normal metal (N_so) with
intrinsic spin-orbit coupling. We show that in the presence of Landau
Fermi-liquid interactions the superconducting proximity effect can induce
non-locally a ferromagnetic exchange field in the normal layer, which
disappears above the superconducting transition temperature of the structure.
The internal Landau Fermi-liquid exchange field leads to the onset of a spin
supercurrent associated with the generation of long-range spin-triplet
superconducting correlations in the trilayer. We demonstrate that the magnitude
of the spin supercurrent as well as the induced magnetic order in the N$_{\rm
so}$ layer depends critically on the superconducting proximity effect between
the S layer and the F and N_so layers and the magnitude of the relevant
Landau Fermi-liquid interaction parameter. We investigate the effect of spin
flip processes on this mechanism. Our results demonstrate the crucial role of
Landau Fermi-liquid interaction in combination with spin-orbit coupling for the
creation of spin supercurrent in superconducting spintronics, and give a
possible explanation of a recent experiment utilizing spin-pumping via
ferromagnetic resonance [Jeon et al., Nat. Mat. 17, 499 (2018)].