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The single-user and two-user (multiple-access) Poisson arbitrarily varying channel (AVC) with input and state (peak and average power) constraints, but unlimited in bandwidth, are considered. For both cases, the deterministic and random code capacity with the average probability of error criterion is obtained. In the single-user case, A.D. Wyner's decoder (see ibid., vol.34, p.1449-61, 1988) attains the deterministic-code capacity whereas for the two-user case, a "nearest neighbor" decoder that belongs to the class of β-decoders introduced by I. Csiszar and J. Korner (see Proc. 3rd Czechoslovak-Soviet-Hungarian Seminar on Information Theory, p.25-44, 1980) is shown to attain the deterministic-code capacity region as claimed.