I'm trying to figure out how to buy an annuity in Germany. Specifically, I'm interested in a fixed immediate life annuity. That is, a product that would offer a fixed payout every month for the lifetime of the owner.
In the US, these products are offered by many financial companies including life insurance companies and investment companies, but I'm not sure what to search for as an equivalent product in Germany.
When I switch languages on Wikipedia from English Annuity(European) to the German equivalent page, I land on a list of pages related to retirement. Life Annuity is linked to Leibrente but this product seems to be tied to real estate, more similar to a reverse mortgage. There is also a false-friend Annuität, which if defined on Wikipedia like Annuity but when I search google, returns results about paying back credits.
Frustrating the search process further, the standard retirement plan in Germany is an annuity, which means that google mostly returns results about the government supported retirement. I want to know how to buy a private additional annuity on top of standard "Rente".
German is my second language, so please jump in and clarify any misunderstandings I may have included in the body of the question due to misreading the Wikipedia pages.