  We study the flat geometry of the least degenerate singularity of a singular
surface in $\mathbb R^4$, the $I_{1}$ singularity parametrised by
$(x,y)\mapsto(x,xy,y^{2},y^{3})$. This singularity appears generically when
projecting a regular surface in $\mathbb R^5$ orthogonally to $\mathbb R^4$
along a tangent direction. We obtain a generic normal form for $I_1$ invariant
under diffeomorphisms in the source and isometries in the target. We then
consider the contact with hyperplanes by classifying submersions which preserve
the image of $I_1$. The main tool is the study of the singularities of the
height function.
