The ISS-based CALET (CALorimetric Electron Telescope) detector can play an important role in indirect search for Dark Matter (DM), measuring the electron+positron flux in the TeV region for the first time directly. With its fine energy resolution of approximately $2%$ and good proton rejection ratio ($1:10^5$) it has the potential to search for fine structures in the Cosmic Ray (CR) electron spectrum. In this context we discuss the ability of CALET to discern between signals originating from astrophysical sources and DM decay or annihilation. We fit a parametrization of the local interstellar electron and positron spectra to current measurements, with either a pulsar or 3-body decay of fermionic DM as the extra source causing the positron excess. The expected CALET data for scenarios in which DM decay explains the excess are calculated and analyzed. The signal from this particular 3-body DM decay which can explain the recent measurements of from the AMS$-02$ experiment is shown to be distinguishable from a single pulsar source causing the positron excess by 5 years of observation with CALET, based on the shape of the spectrum. We also study the constraints from extra-galactic diffuse $gamma$-ray data on this DM-only explanation of the positron excess and show that especially for the possibly remaining parameter space a clearly identifiable signature in the CR electron spectrum exists.
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