Aims. Search for Very High Energy gamma-ray emission in the Kookaburra complex through observations with the H.E.S.S. array. Methods. Stereoscopic imaging of Cherenkov light emission of the gamma-ray showers in the atmosphere is used for the reconstruction and selection of the events to search for gamma-ray signals. Their spectrum is derived by a forward-folding maximum likelihood fit. Results. Two extended gamma-ray sources with an angular (68%) radius of 3.3-3.4 are discovered at high (>13sigma) statistical significance: HESS J1420-607 and HESS J1418-609. They exhibit a flux above 1 TeV of (2.97+/-0.18stat +/-0.60sys)x10-12 and (2.17+/-0.17stat +/-0.43sys)x10-12 cm-2 s-1, respectively, and similar hard photon indices ~2.2. Multi-wavelength comparisons show spatial coincidence with the wings of the Kookaburra. Two pulsar wind nebulae candidates, K3/PSR J1420-6048 and the Rabbit, lie on the edge of the H.E.S.S. sources. Conclusions. The two new sources confirm the non-thermal nature of at least parts of the two radio wings which overlap with the gamma-ray emission and establish their connection with the two X-ray pulsar wind nebulae candidates. Given the large point spread function of EGRET, the unidentified source(s) 3EG J1420-6038/GeV J1417-6100 could possibly be related to either or both H.E.S.S. sources. The most likely explanation for the Very High Energy gamma-rays discovered by H.E.S.S. is inverse Compton emission of accelerated electrons on the Cosmic Microwave Background near the two candidate pulsar wind nebulae, K3/PSR J1420-6048 and the Rabbit. Two scenarios which could lead to the observed large (~10 pc) offset-nebula type morphologies are briefly discussed.
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