The EXTraS project to explore the X-ray Transient and variable Sky searches for coherent signals in the X-ray archival data of XMM-Newton. XMM-Newton performed more than 400 pointed observations in the region of the Large Magellanic Cloud (LMC). We inspected the results of the EXTraS period search to systematically look for new X-ray pulsators in our neighbour galaxy. We analysed the XMM-Newton observations of two sources from the 3XMM catalogue which show significant signals for coherent pulsations. 3XMM J051259.8-682640 was detected as source with hard X-ray spectrum in two XMM-Newton observations, revealing a periodic modulation of the X-ray flux with 956~s. As optical counterpart we identify an early-type star with Halpha emission. The OGLE I-band light curve exhibits a regular pattern with three brightness dips which mark a period of ~1350 d. The X-ray spectrum of 3XMM J051034.6-670353 is dominated by a super-soft blackbody-like emission component (kT ~ 70 eV) which is modulated by nearly 100% with a period of ~1418 s. From GROND observations we suggest a star with r = 20.9 mag as possible counterpart of the X-ray source. 3XMM J051259.8-682640 is confirmed as a new Be/X-ray binary pulsar in the Large Magellanic Cloud. We discuss the long-term optical period as likely orbital period which would be the longest known from a high-mass X-ray binary. The spectral and temporal properties of the super-soft source 3XMM J051034.6-670353 are very similar to those of RX J0806.3+1527 and RX J1914.4+2456 suggesting that it belongs to the class of double-degenerate polars and is located in our Galaxy rather than in the LMC.
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