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//A 2 corner twist in U + 3-cycle of midges in R
//Created with a commutator consisting of Sune as X and R' as Y in commutator [X,Y] with 5 conjugates

U' R2 //outer conjugates
R U' R' //cyclic shift conjugates
R U R' U R U2 R' U2 //Sune = X
R' //Y
U2 R U2 R' U' R U' R' //Inverse of Sune = X'
R //=Y'
R U R' //inverse of cyclic shift conjugates
R2 U //inverse of outer conjugates

//A 2 corner twist in U + 3-cycle of midges in R
//Created with a commutator consisting of Sune as X and R' as Y in commutator [X,Y] with 5 conjugates

U' R2 //outer conjugates
R U' R' //cyclic shift conjugates
R U R' U R U2 R' U2 //Sune = X
R' //Y
U2 R U2 R' U' R U' R' //Inverse of Sune = X'
R //=Y'
R U R' //inverse of cyclic shift conjugates
R2 U //inverse of outer conjugates

U2 R U2 R’ U R’ U’ // Square
R2 U’ R U2 R’ U’ R // RB Pair
U2 R U2 R2 U’ R2 U’ R2 U2 // H OCLL
R’ U R U R’ U’ R’ U’ R’ U R’ U // Ua-Perm
35 HTM, pretty “standard” solution

Next: R2 U’ R’ L D R2 D' L' U' F2 U L2 D R2 D2 B2 D L2 U2 R2 U’

This is solution is nearly the same as my previous, but I wasn't fortunate with the 3-cycle of midges AND the two corner twist both going in the needed direction, when Sune was used to make a commutator which twists two corners and does a 3-cycle of midges. So I forced it, but it costed extra moves. (It's ironic, but when I made the previous, I knew I "got lucky" with this, and on the very next solution, I got "unlucky". LOL.)

U R U2 R2 U R2 U R U' R' //Premoves

R' U R2 U R2 U' R' //Setup to change direction of 3-cycle of midges
R U R' U R U2 R' U2 //Sune = X
R //Y
U2 R U2 R' U' R U' R' //X'
R'//Y'
R U R2 U' R2 U' R//undo setup

Next: U2 B' R2 B R2 U R U' R' B U' B' R' U' R U' R

EDIT:

OFF TOPIC

For those who are interested in this thread, perhaps you would be interested in 2-Gen parity algorithms for the 4x4x4.

If so, and you are not aware, see this post which contains my 2-Gen video series that I uploaded last year.

In addition, I never made a video explanation of the following two (nearly <M,U> 2-Gen) parity algorithms, but if someone would like me to explain them in a video, I can.

Single Parity
r U' M' U M' U' M' U' M Uw2 r Uw2 U' M' U2 M' U M U2 M U M' U M U M' U' l' (31,28)

Double Parity
r' U M' U' M' U M' U M Uw2 l' Uw2 M' U M U' M' U M' U M U2 M' U2 r (29,25)