  We study the mathematical properties of a model of cell division structured
by two variables, the size and the size increment, in the case of a linear
growth rate and a self-similar fragmentation kernel. We first show that one can
construct a solution to the related two dimensional eigenproblem associated to
the eigenvalue 1 from a solution of a certain one dimensional fixed point
problem. Then we prove the existence and uniqueness of this fixed point in the
appropriate $\mathrm{L}^1$ weighted space under general hypotheses on the
division rate. Knowing such an eigenfunction proves useful as a first step in
studying the long time asymptotic behaviour of the Cauchy problem.
