A large portion of the 2GLLs and diag ZBLLs can be solved with a conjugated sexy move (or sledgehammer, which amounts to the same thing).

The easiest example is Sune: R U R' U R U2 R' = [R U R' : [U, R]]. Another one is [r U2 R' : [R', F]] (one of the common diag-L COLL algs; rearranging this a bit gives the x R2 F R F' Ja perm alg that a few people use). Looking outside of ZBLL, we also have [F : [R, U]] and [R' U' : [R', F]], among a few others.

Some algs are conjugated single moves, e.g.

[R U' R2 D' r : U2] (from COLL/ZBLL)

[r' D R2 U R' : U2] (from OLLCP)

[R U' R2 B2 D' r : U2] (an optimal N perm alg)

[r' D B2 R2 U R' : U2] (an optimal 4-flip N perm alg)

[F U' R2 D R' : U2] (from OLLCP/flipped line 1LLL)

[r u' R2 u r' : U2] = [L B' D2 B L' : U2] (optimal corner twist alg)

[r U' M2 U r' : U2] (from ELL)

[R' U2 r U' r' : U2] (from OLL)

[M' U' M U' M' : U2] (from ELL)

[F R U' R' : U'] and inverse (from OLL/OLLCP)

These all use a bunch of moves to shuffle the top layer pieces (messing up F2L along the way), do a U/U2/U' move to permute them, then restore F2L. It's theoretically possible to get algs for most (unfortunately, not all) 1LLL cases this way, although it goes without saying that such algs aren't necessarily *good* (e.g. that N perm listed above).

Some algs can be decomposed into a simpler alg with inserted commutators, e.g. the RUD R perms: R U2 R D R' U R D' R' U' R' U R U R' = [R U2 : @ [R', U]] with @ = [R D R', U].