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RR Lyrae variable stars are the primary Population II distance indicator. Likewise, the Large Magellanic Cloud (LMC) constitutes a key step in the extragalactic distance scale. By combining near-IR photometry and spectroscopically measured metallicities for a homogeneous sample of 50 RR Lyr stars in the LMC, we investigate the metallicity dependence of the period-luminosity relation in the near-infrared (IR), use the newly derived relations to re-derive the distance to the LMC, and compare the distance moduli obtained from RR Lyr and red clump stars. This paper presents new (single-epoch) J-band and (multi-epoch) K_s-band photometry of RR Lyr stars in 7 different LMC fields, observed with the near-IR camera SOFI at ESO's New Technology Telescope. Additional K_s-band data for another two LMC fields were taken with the ISPI infrared array at CTIO's Blanco 4m telescope. The near-IR photometry was cross-correlated with the MACHO and OGLE databases, resulting in a catalog of 62 RR Lyr stars with BVRIJK_s photometry. A subsample of 50 stars also has spectroscopically measured metallicities. In the deep JK color-magnitude diagrams of 7 fields, red giant branch, red clump and RR Lyr stars are detected. The majority of RR Lyr stars are located within the instability strip with near-IR colors between 0.14≤(J-K_s)₀<0.32. The period-luminosity relation only has a very mild dependence on metallicity in the K band, consistent with no dependence: M_Ks=2.11(±0.17)logP+0.05(±0.07)[Fe/H]-1.05. In the J band the currently available data do not allow firm conclusions regarding the metallicity dependence of the period-luminosity relation. The distance modulus of the LMC, derived using our near-IR period-luminosity-metallicity relation for RR Lyr stars, is (m-M)₀=18.53±0.13, in very good agreement with the distance modulus from the red clump stars, 18.46±0.07. However, LMC modulus derived from the RR Lyrae stars depends on the parallax of the star RR Lyrae.