This paper considers the Cauchy problem for the following systems of Broadwell's type When the nonlinear function F takes the special form f 1 f 2 -f 3 2 , (1) is a simple mathematical model of gas kinetics, the so called Broadwell model [1] (see also [2], [6] [13], [16], [17], [18] and the references therein). It describes an idealization of a discrete velocity gas of particles in one dimension subject to a simple binary collision mechanism. Let ρ = f 1 + f 2 + 4f 3 , m = f 1 - f 2 , s = f 3 . (1) may be written as follows: ρ t + m x = 0 m t + (ρ - 4s) x = 0 (2) The conditions for (ρ, m, s) to be a local Maxwellian are s = 1/6(2 √1+3m 2 /ρ 2 )ρ (3) for the Broadwell case F = f 1 f 2 - f 3 2 . If the nonlinear collision function F can be written as F = h 1 (f 3 ) - h 2 (f 1 + f 2 + 4f 3 ) for some nonlinear functions h, h 1 and h 2 the conditions on (ρ, m, s) to be a local Maxwellian are s = h(ρ). (4) The equilibrium systems corresponding to (3) and (4) are the following Euler equations (5) and p-system (6): ρt+ m x = 0 (5) m t + (ρG(u)) x = 0.