  In this paper, we prove local existence and uniqueness of analytic
sharp-front solutions to a generalised SQG equation by the use of an abstract
Cauchy--Kowalevskaya theorem. Here, the velocity is determined by $u =
|\nabla|^{-2\beta}\nabla^\perp\theta $ which (for $1<\beta\leq 2$) is more
singular than in SQG. This is achieved despite the appearance of
pseudodifferential operators of order higher than one in our equation, by
recasting our equation in a suitable integral form. We also provide a full
proof of the abstract version of the Cauchy--Kowalevskaya theorem we use.
