In the weeks prior to the visit, Kenny declared that he would "effectively and conclusively" deal with the issue of his departure from office following his return to Dublin. A long-running scandal over a police whistle-blower has undermined Kenny's leadership of the Fine Gael party and opened fissures within the ruling coalition.
Several independent members of parliament, including Katherine Zappone, could lose their seats if a snap election is called.
Dublin could seek guarantees from the UK government in Brexit talks allowing for future Irish reunification.
Sinn Fein will likely increase their numbers in a forthcoming poll, but are unlikely to form part of the next administration.
Renewed divisions over domestic water charges could prompt the downfall of the current government.
