  The Symmetries of Feynman Integrals (SFI) method is extended for the first
time to incorporate an irreducible numerator. This is done in the context of
the so-called vacuum and propagator seagull diagrams, which have 3 and 2 loops,
respectively, and both have a single irreducible numerator. For this purpose,
an extended version of SFI (xSFI) is developed. For the seagull diagrams with
general masses, the SFI equation system is found to extend by two additional
equations. The first is a recursion equation in the numerator power, which has
an alternative form as a differential equation for the generating function. The
second equation applies only to the propagator seagull and does not involve the
numerator. We solve the equation system in two cases: over the singular locus
and in a certain 3 scale sector where we obtain novel closed-form evaluations
and epsilon expansions, thereby extending previous results for the
numerator-free case.
