Original post
I've thought about a way to do LL + LS quite quickly. My idea is to orient and permute the corners while inserting LS, then there's 4 OLLs you can get from there(one of them being OLL solved completely). Then use ELL to orient and permute the LL edges. That would be LS, OLL and PLL solved in 2 algorithms.
I just can't find a method that orients and permutes the corners while inserting LS. If anyboody know of such methods please tell me.
It would also be great if somebody could teach me how you find all the possible cases. E.g. how would you go about finding all the possible cases for PLL.
This is if there isn't any method out there that meets my requirements, I will maybe be able to make it myself.
2nd post
I have done some work and I am pretty sure, that for a method that orients and permutes the LL corners while inserting LS, while egnoring LL edges, there wil be 324 unique cases. I haven't done the math, since i don't know how, therefore I am not sure if there is only 324 cases. If anyone could do the math i would really appreciate it.
I did this by using Cube Explorer and making it figure out a lot of generators for cases where the LS corner and edge are paired up and in LL right about the slot they're supposed to go in to. I let it go on to find about 130.000 generators, out of those 130.000 there were 324 unique cases. But I'm still not sure if there are more.
I think, but I'm not sure that i am the first to do this method, I was thinking of calling it HVLS(Hansen Vejlø Last Slot((Hansen for the guy whom helped me make this method and Vejlø for myself)). An alternative name would be COPLS(Corners Permutation and Orientation Last Slot), OPCLS(Orientation and Permutation of Corners Last Slot) or CLLLS(Corners Last Layer Last Slot).
If I'm not the first to come up with this, please feel free to correct me, i will gladly credit him/her.
I will be learning these 324 algorithms over this and next year(I'm not that fast a learning algorithms, so it'll take a while), if there is found more cases I will certainly learn those as well. I hope this will become a replacement for the normal LS + OLL + PLL, though the Alg. count is quite high.