  We investigate a functional equation which resembles the functional equation
for the generating function of a lattice walk model for the quarter plane. The
interesting feature of this equation is that its orbit sum is zero while its
solution is not algebraic. The solution can be interpreted as the generating
function of lattice walks in $\mathbb{Z}^2$ starting at $(-1,-1)$ and subject
to the restriction that the coordinate axes can be crossed only in one
direction. We also consider certain variants of the equation, all of which seem
to have transcendental solutions. In one case, the solution is perhaps not even
D-finite.
