  The aim of this work is focused on the investigation of the algebraic
complete integrability (a.c.i) of the Toda lattice associated with the twisted
affine Lie algebra (a_4^{\left(2\right)}). First, we prove that the generic
fiber of the momentum map for this system is an affine part of an abelian
surface. Second, we show that the flows of integrable vector fields on this
surface are linear. Finally, using the formal Laurent solutions of the system,
we provide a detailed geometric description of these abelian surfaces and the
divisor at infinity.
