AT...
As AR says, your post is misleading with the phrase "failed engine". This covers many scenarios, but my broad interpratation of "failed engine" would be one that wouldn't run at all
Certain problems with the engine might mean it can be run at idle, or low power. As AR says, this keeps certain ancillary systems going, so makes life easier for the FC. Of course, Engineering might not thank you if it leads to further damage, but that's their problem
However, certainly on types I have flown / company I work for, the engine is to be treated as "failed". So on a twin, it's a Land ASAP - so your statement "in order to continue the flight" is incorrect. Why? Well, if it will only run at idle, it's not a lot of use for approach or GA... and if the remaining engine failed you have no power.
As to what failures / aircraft / driftdown etc. consult your Airline / Aircraft Ops Manuals - I am assuming you are a pilot
If you are not, the answers are endless and variable, and best left at that...