  The extended Linear Sigma Model (eLSM) is an effective hadronic model based
on the linear realization of chiral symmetry $SU(N_f)_L \times SU(N_f)_R$, with
(pseudo)scalar and (axial-)vector mesons as degrees of freedom. In this paper,
we study the low-energy limit of the eLSM for $N_f=2$ flavors by integrating
out all fields except for the pions, the (pseudo-)Nambu--Goldstone bosons of
chiral symmetry breaking. We only keep terms entering at tree level and up to
fourth order in powers of derivatives of the pion fields. Up to this order,
there are four low-energy coupling constants in the resulting low-energy
effective action. We show that the latter is formally identical to Chiral
Perturbation Theory (ChPT), after choosing a representative for the coset space
generated by chiral symmetry breaking and expanding up to fourth order in
powers of derivatives of the pion fields. Two of the low-energy coupling
constants of the eLSM are uniquely determined by a fit to hadron masses and
decay widths. We find that their tree-level values are in reasonable agreement
with the corresponding low-energy coupling constants of ChPT. The other two
low-energy coupling constants are functions of parameters that can in principle
be determined by $\pi\pi$ scattering, which has not yet been studied within the
eLSM. Therefore, we use the respective values from ChPT to make a prediction
for the values of these parameters in the eLSM Lagrangian.
