Normalization flow in the presence of a resonance

Following [Tre23], we develop an approach to the Hamiltonian theory of normal forms based on continuous averaging. We concentrate on the case of normal forms near an elliptic singular point, but unlike [Tre23] we do not assume that frequences of the linearized system are nonresonant. We study analytic properties of the normalization procedure. In particular we show that in the case of a codimension one resonance an analytic Hamiltonian function may be reduced to a normal form up to an exponentially small reminder with explicit estimates of the reminder and the analyticity domain.