  In this paper we consider generalised Proca theories coupled to any
background field and with time-time and time-space components of Hessian of the
vector sector are zero, whereas the space-space part is non-degenerate. By
using Faddeev-Jackiw analysis, we derive the conditions that these theories
have to satisfy in order for the vector sector to have three propagating
degrees of freedom. Most of these conditions are trivialised due to
diffeomorphism invariance requirements. This leaves only a condition that a
complicated combination of terms should not be trivially zero. This condition
is therefore easy to be fulfilled. For completeness, we have also investigated
on how diffeomorphism invariance helps in simplifying Faddeev-Jackiw brackets.
