Nucleosynthetic yields of Z=$10^{-5}$ intermediate-mass stars

Abridged: Observed abundances of extremely metal-poor (EMP) stars in the Halo hold clues for the understanding of the ancient universe. Interpreting these clues requires theoretical stellar models at the low-Z regime. We provide the nucleosynthetic yields of intermediate-mass Z=$10^{-5}$ stars between 3 and 7.5 $M_{sun}$, and quantify the effects of the uncertain wind rates. We expect these yields can be eventually used to assess the contribution to the chemical inventory of the early universe, and to help interpret abundances of selected C-enhanced EMP stars. By comparing our models and other existing in the literature, we explore evolutionary and nucleosynthetic trends with wind prescriptions and with initial metallicity. We compare our results to observations of CEMP-s stars belonging to the Halo. The yields of intermediate-mass EMP stars reflect the effects of very deep second dredge-up (for the most massive models), superimposed with the combined signatures of hot-bottom burning and third dredge-up. We confirm the reported trend that models with initial metallicity Z$_{ini}$ <= 0.001 give positive yields of $^{12}C, ^{15}N, ^{16}O$, and $^{26}Mg$. The $^{20}Ne, ^{21}Ne$, and $^{24}Mg$ yields, which were reported to be negative at Z$_{ini}$ = 0.0001, become positive for Z=$10^{-5}$. The results using two different prescriptions for mass-loss rates differ widely in terms of the duration of the thermally-pulsing (Super) AGB phase, overall efficiency of the third dredge-up episode, and nucleosynthetic yields. The most efficient of the standard wind rates frequently used in the literature seems to favour agreement between our yield results and observational data. Regardless of the wind prescription, all our models become N-enhanced EMP stars.