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In this paper we consider the following Toda system of equations on a compact surface: Here h 1 , h 2 are smooth positive functions and ρ 1 , ρ 2 two positive parameters. In this note we compute the Leray-Schauder degree mod Z 2 of the problem for ρ i ∈ (4πk, 4π(k + 1)) (k ∈ N). Our main tool is a theorem of Krasnoselskii and Zabreiko on the degree of maps symmetric with respect to a subspace. This result yields new existence results as well as a new proof of previous results in literature.doi:10.1090/s0002-9939-2015-12484-7 fatcat:s6wunzpojja3lb2drs2fpeqt7u