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The NA48/2 experiment at CERN reports the first observation of the $K^{\pm}\rightarrow \pi^{0} \pi^{0} \mu^{\pm} \nu$ decay based on a sample of 2437candidates with 15% background contamination collected in 2003--2004. The decaybranching ratio in the kinematic region of the squared dilepton mass above$0.03$~GeV$^2/c^4$ is measured to be $(0.65 \pm 0.03) \times 10^{-6}$. Theextrapolation to the full kinematic space, using a specific model, is found tobe $(3.45 \pm 0.16) \times 10^{-6}$, in agreement with chiral perturbationtheory predictions.

First observation and study of the $K^{\pm} \rightarrow \pi^{0} \pi^{0} \mu^{\pm} \nu$ decay