The KLM+KLN Auger electron spectrum of rubidium (Z=37) emitted in the electron capture decay of radioactive $^{83}$Sr in a polycrystalline platinum matrix and also $^{85}$Sr in polycrystalline platinum and carbon matrices as well as in an evaporated layer onto a carbon backing was experimentally studied in detail for the first time using a combined electrostatic electron spectrometer. Energies, relative intensities, and natural widths of fifteen basic spectrum components were determined and compared with both theoretical predictions and experimental data for krypton (Z=36). Relative spectrum line energies obtained from the semi-empirical calculations in intermediate coupling scheme were found to agree within 3$sigma$ with the measured values while disagreement with experiment exceeding 3$sigma$ was often observed for values obtained from our multiconfiguration Dirac-Hartree-Fock calculations. The absolute energy of the dominant spectrum component given by the semi-empirical approach agrees within 1$sigma$ with the measured value. Shifts of + (0.2$pm$0.2) and - (1.9$pm$0.2) eV were measured for the dominant KLM spectrum components between the $^{85}$Sr sources prepared by vacuum evaporation on and implanted into the carbon foil, respectively, relative to $^{85}$Sr implanted into the platinum foil. A value of (713$pm$2) eV was determined for the energy difference of the dominant components of the KLM+KLN Auger electron spectra of rubidium and krypton generated in the polycrystalline platinum matrix. From the detailed analysis of the measured data and available theoretical results, the general conclusion can be drawn that the proper description of the KLM+KLN Auger electron spectrum for Z around 37 should still be based on intermediate coupling of angular momenta taking into account relativistic effects.
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