The authors focus on a class of so-called standard (reversible) models of small \(C^\infty\)-perturbations of four-dimensional discontinuous vector fields \[ Z_{a,b,c}(x_1,x_2,x_3,x_4) = L_a(x_1,x_2,x_3,x_4) + P(x_1,x_2,x_3,x_4), \]
\[ L_a(x_1,x_2,x_3,x_4) = x_2 (\partial /\partial x_1) + x_3 (\partial /\partial x_2) + x_4 (\partial /\partial x_3) + a\; \text{sgn}(x_1) (\partial /\partial x_4), \] where \(P\) is a (given) polynomial and \(a= \pm 1\). Using a particular example first, they provide conditions under which \(Z_{a,b,c}\) present one-parameter families of one-periodic orbits terminating at the origin, provide an exact algorithm to find these orbits, study their local stability and ways in which they disappear, and show how the results can be extended to any small polynomial perturbations of \(L_a\).