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This paper is devoted to the study of interactions between stationary electromagnetic sources for the minimal and nonminimal CPT-odd photon sector of the Standard Model Extension (SME), where we search mainly for physical phenomena not present in the Maxwell electrodynamics. First we consider the minimal CPT-odd sector, where the Lorentz violation is caused by the Carroll-Field-Jackiw (CFJ) term, namely $\simε^{μναβ}\left(k_{AF}\right)_μA_νF_{αβ}$, and we treat the Lorentz breaking parameter $\left(k_{AF}\right)^μ$ perturbatively up to second order. We consider effects due to the presence of point-like charges, Dirac strings and point-like dipoles. In special, we calculate the electromagnetic field produced outside the string and investigate the so called Aharonov-Bohm bound states in Lorentz violation context. After, we consider a model where the Lorentz violation is generated by the higher-derivative version of the CFJ model, namely $\simε^{μναβ}V_μA_ν\Box F_{αβ}$, which is a dimension five term of the CPT-odd sector of the nonminimal SME. For this higher-derivative model, we obtain effects up to second order in $V^μ$ related to the presence of point-like charges and a steady current line. We use overestimated constrains for the Lorentz violation parameters in order to investigate the physical relevance of some results found in atomic systems. We also make an overestimate for the background vectors using experimental data from the atomic electric field.