The aurorae of Uranus were recently detected in the far ultraviolet with the Hubble Space Telescope (HST) providing a new, so far unique, means to remotely study the asymmetric Uranian magnetosphere from Earth. We analyze here two new HST Uranus campaigns executed in Sept. 2012 and Nov. 2014 with different temporal coverage and under variable solar wind conditions numerically predicted by three different MHD codes. Overall, HST images taken with the Space Telescope Imaging Spectrograph reveal auroral emissions in three pairs of successive images (one pair acquired in 2012 and two in 2014), hence six additional auroral detections in total, including the most intense Uranian aurorae ever seen with HST. The detected emissions occur close the expected arrival of interplanetary shocks. They appear as extended spots at southern latitudes, rotating with the planet. They radiate 5-24 kR and 1.3-8.8 GW of ultraviolet emission from H2, last for tens of minutes and vary on timescales down to a few seconds. Fitting the 2014 observations with model auroral ovals constrains the longitude of the southern (northern) magnetic pole to 104 ± 26∘(284±26∘) in the Uranian Longitude System. We suggest that the Uranian near-equinoctial aurorae are pulsed cusp emissions possibly triggered by large-scale magnetospheric compressions.