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The rest-frame near-IR spectra of intermediate age (1-2Gyr) stellar populations are dominated by carbon based absorption features offering a wealth of information. Yet, spectral libraries that include the near-IR wavelength range do not sample a sufficiently broad range of ages and metallicities to allow for accurate calibration of stellar population models and thus the interpretation of the observations. In this paper we investigate the integrated J- and H-band spectra of six intermediate age and old globular clusters in the Large Magellanic Cloud (LMC). The observations for six clusters were obtained with the SINFONI integral field spectrograph at the ESO VLT Yepun telescope, covering the J (1.09-1.41µm) and H-band (1.43-1.86µm) spectral range. The spectral resolution is 6.7Å in J and 6.6Å in H-band (FWHM). The observations were made in natural seeing, covering the central 24''x24'' of each cluster and in addition sampling the brightest eight red giant branch and asymptotic giant branch (AGB) star candidates within the clusters' tidal radii. Targeted clusters cover the ages of ∼1.3Gyr (NGC1806, NGC2162), 2Gyr (NGC2173) and ∼13Gyr (NGC1754, NGC2005, NGC2019). H-band C₂ and K-band ¹²CO(2-0) feature strengths for the LMC globular clusters are compared to the models of Maraston (2005). C₂ is reasonably well reproduced by the models at all ages, while ¹²CO(2-0) shows good agreement for older (age≥2Gyr) populations, but the younger (1.3Gyr) globular clusters do not follow the models. We argue that this is due to the fact that the empirical calibration of the models relies on only a few Milky Way carbon star spectra, which show different ¹²CO(2-0) index strengths than the LMC stars. The C₂ absorption feature strength correlates strongly with age. It is present essentially only in populations that have 1-2Gyr old stars, while its value is consistent with zero for older populations. The distinct spectral energy distribution observed for the intermediate age globular clusters in the J- and H-bands agrees well with the model predictions of Maraston for the contribution from the thermally pulsing AGB phase. In this pilot project we present an empirical library of six LMC globular cluster integrated near-IR spectra that are useful for testing stellar population models in this wavelength regime. We show that the H-band C₂ absorption feature and the J-, H-band spectral shape can be used as an age indicator for intermediate age stellar populations in integrated spectra of star clusters and galaxies.