  Motivated by the relation between Schur algebra and the group algebra of a
symmetric group, along with other similar examples in algebraic Lie theory, Min
Fang and Steffen Koenig addressed some behaviour of the endomorphism algebra of
a generator over a symmetric algebra, which they called gendo-symmetric
algebra. Continuing this line of works, we classify in this article the
representation-finite gendo-symmetric algebras that have at most one
isomorphism class of indecomposable non-injective projective module. We also
determine their almost {\nu}-stable derived equivalence classes in the sense of
Wei Hu and Changchang Xi. It turns out that a representative can be chosen as
the quotient of a representation-finite symmetric algebra by the socle of a
certain indecomposable projective module.
