What came to me in my nap was the question: where all can I meaningfully insert 'ala' in the framework given (assuming no double negations for now):
(ona (wile(tawa tomo))). "they want to go home" (these placements are logical, not necessarily linear -- though most are, somehow).
The answer seems to be:
1 (ona (wile(tawa (tomo ala)))) "They want to go to no home" "to someplace other than a home"
2 (ona (wile((tawa tomo)ala))) "They want to not go home" "something other than go home"
3 (ona (wile((tawa ala) tomo))) "They want to does something to homes other than go to them."
4 (ona ((wile(tawa tomo))ala)) "They do something other than want to go home"
5 (ona ((wile ala)(tawa tomo))) "They allow/are indifferent to/don't desire going home"
6 ((ona (wile(tawa tomo))) ala) *They don't want to go home"
7 ((ona ala) (wile(tawa tomo))) "None of them want to go home."
Given a simple comma convention, most of these have easy forms. 1 needs no commas, 2 has a comma before 'ala', 3 needs no comma, nor does 7. 6 cannot be reached with the words in this order using the present rules, since it is sentential negation and that seems to require the 'ala' immediately after the verb (the first word after 'li'). As a result, 5 does not have a distinctive form, since its natural form coincides with the sentential form. Similarly, even with the present suggested comma convention (always optional, often appreciated), 4 can't be distinguished from 2 (I think we can reject the suggestion of double commas out of hand).
There is one further, questionable case:
(ona (((wile tawa) ala) tomo))) "They do something to homes other than want to go to them"
This is distinguishable from 3 by a comma before 'ala', but it also seems to be a different structure. Still, it clearly fits into a similar linear pattern.
I confess I have no idea where all these forms come from transformationally or even generatively, but I note that exactly the same patterns recur meaningfully with both 'kin' and 'taso', which are, alas, as opaque as 'ala'. However, 'kin' clearly comes from several sources, which may be a clue about 'ala' (and 'taso). In particular, some of these 'ala's are pretty clearly polar negations (as shown by "something other than"), others sentential (at some level).
Coming up in all this is the question of sentential negation. Does it really have to go right after the "verb" (first word after 'li') or might it go after a phrase of sort of unitary content, 'pilin pona', say, or 'kama sona'? There are examples, even in authoritative texts, of the latter approach, but the general tendency is toward the narrow rule, even when it looks weird (because, I think, it splits one concept in English) (Since this is figuring out what is going on in tp, not advocating or considering changes, I skip over the notion that sentential negation should go somewhere other than after the verb.)
Hanging on this to some extent is the similar question about the 'x ala x' formula for Y/N questions, which I take to come from 'x ala anu x' at a very superficial level. That is, can the 'x ala x' formula be used with other than the verb -- with things more pertinent to the interests at hand, say. Cases of this variation are much more common than those with simple negations, I think, even occasionally moving outside the verb phrase to the DO or PP (I don't remember a case in the subject, but would be surprised to find one).