  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. These
lines trace the high-density gas, which remains almost entirely unexplored in
z$\geq$1 galaxies. Additionally, these dense-gas tracers are potentially
powerful 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) emission
is clearly detected and resolved. This is the third detection of this line in a
high-redshift star-forming galaxy. We find an unusually high HCO+/HCN intensity
ratio of $\geq$2.2. Based on the photodissociation region modelling, the most
likely explanation for the elevated HCO+/HCN ratio is that SDP.81 has low
mechanical heating - less than 10% of the total energy budget - and a sub-solar
metallicity, Z=0.5 Z$_\odot$. While such conditions might not be representative
of the general population of high-redshift dusty galaxies, lower-than-solar
metallicity might have a significant impact on gas masses inferred from CO
observations. In addition, we report the detection of CO(0-1) absorption from
the foreground lensing galaxy and CO(1-0) emission from a massive companion to
the lensing galaxy, approximately 50 kpc to the southeast.
