Your question is for specialists. This is not an answer, but a sharing. I read Levinas some years ago as well as  Paul Ricoeur (not the piece you mention but Recognition) for my dissertation/book on Recognition. Levina's language is far more obscure than Ricoeur's, I found out. To read Levina, I needed to pause and look back quite often where (what part of the book) I am.  This is not a complaint, ha!

Levinas' book with me is:

Lévinas, E. (1991). Totality and infinity : an essay on exteriority. Dordrecht; Boston; London: Kluwer Academic Publishers.

The term "sollicitude" is not used in this English version, hence, 'care'. In this work, 'care' first appears in the sub-section devoted to truth and justice (which is in turn under the Section I on the Same and the Other).  Levina is up to pre-categorization in this Section.  Care (perhaps "sollicitude") is for being and its existence, Levina's statement such as "priority of care over contemplation" , for me, only confirms his well-known emphasis on ethics (relational discourse) over metaphysics.  The next big section dealing with care starts with his observation on Heidegger.  I think you have the book, so I will not elaborate further on it.

Just a quick observation, if you allow me. The two, Ricoeur's responsibility and Levina's care seem to share in common : (1) Ethically, they are on 'duty towards' oneself or others  (2) They are about 'dues', hence justice, and justice in the Aristotelian/Thomistic  sense, that is, proving what is due to a being and what its existence deserves.    But this is just a hunch since I did not read "soi-même comme un autre". Yet, these French words point to the same coordinates of Levina's "the Same and the Other" . I do not think that these are just a coincidence. Here is something you can grab and go deeper. Happy research

I wrote a long comment. This is odd. For an hour  it was gone ..and it is back now

I don´t think that a sinthesis between the concept of responsability of Levinas and the concept of "sollicitude" of Ricoeurit is not possible. In fact, Lévinas sees responsability tottaly on my side, that is to say that each one of us is totally responsable for the other, in face of the "visage de l'autre". It is "L'autre" that claims for me, for my full attention and responsability before him, in a way that I am his hostage!

Tha answer you request is somehow answered in the way you put the question, because Ricoeur, as you say, strongly criticizes Levinas and his concept of responsability, considering that it isto much excessive and takes responsability to far away... In the other hand, the concept of "solicitude" in Ricoeur's thought necessarily includes reciprocity. As he says in "Soi Mêmme comme un Autre": "...l'estime de soi et la solicitude [les deux concepts constitutifs de la solicitude et qui ensemble permettre de definir la solicitude] ne puissent se vivre et se penser l'une sans l'autre" (p. 212)! Or, reciprocity is that what Lévinas excludes! And "Soi Mêmme comme un Autre" is very clear about that. There, in the "La petite éthique", as Ricoeur's calls it, you can read that  "hapiness" or "Good" is to " Live well with and for others in just institutions"; As Ricoeur writes: ""Appelons la visée éthique la visée de la vie bonne avec et pour autrui dans des institutions justes" (p. 202). And it is no coincidence that friendship is the  the paradigmatic exemple of  "sollicitude" for Ricoeur.

So, I don´t think it is possible to make a sinthesis between the concept of responsability of Levinas and the concept of "sollicitude" of Ricoeur, and more: I do not see what advantage it can have. On the contrary, it would dilute the differences between Ricoeur and lévinas...

@Maria

1. Is 'advantage' the raison d'tre of philosophical inquiries?

2. I do not think that  total synthesis of any two thinkers is ever possible. We should probably  give up that dream. If so, why should we demonize a partial synthesis or comparison of two ideas? Vive la liberté