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We present deep ALMA Band 3 observations of the HCN, HCO+, and HNC (4-3)emission in SDP.81, a well-studied z = 3.042 strongly lensed galaxy. Theselines trace the high-density gas, which remains almost entirely unexplored inz$\geq$1 galaxies. Additionally, these dense-gas tracers are potentiallypowerful diagnostics of the mechanical heating of the interstellar medium.While the HCN(4-3) and HNC(4-3) lines are not detected, the HCO+(4-3) emissionis clearly detected and resolved. This is the third detection of this line in ahigh-redshift star-forming galaxy. We find an unusually high HCO+/HCN intensityratio of $\geq$2.2. Based on the photodissociation region modelling, the mostlikely explanation for the elevated HCO+/HCN ratio is that SDP.81 has lowmechanical heating - less than 10% of the total energy budget - and a sub-solarmetallicity, Z=0.5 Z$_\odot$. While such conditions might not be representativeof the general population of high-redshift dusty galaxies, lower-than-solarmetallicity might have a significant impact on gas masses inferred from COobservations. In addition, we report the detection of CO(0-1) absorption fromthe foreground lensing galaxy and CO(1-0) emission from a massive companion tothe lensing galaxy, approximately 50 kpc to the southeast.

PRUSSIC II -- ALMA imaging of dense-gas tracers in SDP.81: Evidence for low mechanical heating and a sub-solar metallicity in a z=3.04 dusty galaxy