Classification of symmetric special biserial algebras with at most one non-uniserial indecomposable projective

We consider a natural generalisation of symmetric Nakayama algebras, namely, symmetric special biserial algebras with at most one non-uniserial indecomposable projective module. We describe the basic algebras explicitly by quiver and relations, then classify them up to derived equivalence and up to stable equivalence of Morita type. This includes the algebras of [Bocian-Holm-Skowro\'nski, J. Pure Appl. Algebra 2004], where they study the weakly symmetric algebras of Euclidean type, as well as some algebras of dihedral type.
Comment: Changes in the third (this) version: One reference has been added. Paper to appear in the Proceedings of the Edinburgh Mathematical Society. Changes in the second version: The classification up to stable equivalence of Morita type has been added and a few references added