In this paper, several boundary value problems for second order complex partial differential systems of bi-analytic functions were investigated on the bicylinder. Homogeneous and nonhomogeneous Dirichlet problems for $ (\lambda, 1) $ bi-analytic functions were first discussed on the bicylinder. Applying the Cauchy-Pompeiu formula and the properties of the Poisson kernel, the expressions of the solutions to the Dirichlet problems were obtained. Thereafter, the Riemann problems and the inverse problems for $ (\lambda, 1) $ bi-analytic functions were explored on the generalized bicylinder in $ \mathbb{C}^2 $. Applying the Plemelj formula, the solutions to the corresponding problems were obtained.