What are the French verbs that have invariable past participles?

Question

I am currently researching French verbs, and have found some interesting patterns.

For first group verbs such as aimer, the forms of the past participle are "-é" (masculine singular), "–ée" (feminine singular), "–és" (masculine plural), and "–ées" (feminine plural).

For second group verbs such as aplanir, the forms of the past participle are "-i" (masculine singular), "-ie" (feminine singular), "-is" (masculine plural), and "-ies" (feminine plural).

Even though they are irregular and vary quite a bit, for third group verbs such as appendre, the forms of the past participle are "–u" (masculine singular), "-ue" (feminine singular), "-us" (masculine plural), and "-ues" (feminine plural).

When it comes to the past participle of French verbs, choosing the right form is important in the following circumstances.

1. The direct object of a sentence comes before "avoir + (past participle)"
2. "être + (past participle)"
3. The "être + (past participle)" form of a reflexive verb when there is no direct object or when the direct object comes before the reflexive verb
4. The past participle forms the passive voice
5. The past participle acts as an adjective

Having said that, I found out that there are French verbs that have invariable past participles. That is, the past participle does not change its form at all.

For example, "être" has the past participle "été", which Wiktionary claims does not change form because it is intransitive. However, it also claims that "été" had other forms in older writings.

So, if you know any French verbs that have invariable past participles, please tell me.

Answer 1

All verbs for which no past participle agreement is possible have, by definition, an invariable past participle.

Être is one of them because it is an intransitive verb (it never has any complement, it only has predicate nominatives, attributs du sujet). No agreement is possible with intransitive verbs simply because there is no direct object to agree with.

This page (BDL - OQLF) lists hundreds of invariable past participles:

Le participe passé des verbes intransitifs, des verbes transitifs indirects et des verbes impersonnels est toujours invariable, puisque ces verbes n’ont pas de complément direct avec lequel s’accorder.
Participes passés toujours invariables
abouti
accédé
achoppé
acquiescé
adhéré
afflué
agi
agonisé
amerri
[...]

Note: In rare cases, past participles of these verbs used adjectivally might nevertheless be found with a agreement like in Cette maison est accédée par une grande porte cochère instead of the expected Cette maison à laquelle on accède par une grande porte cochère or Cette maison accessible par une grande porte cochère. We still can't say J'accède cette maison par une grande porte cochère though.