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A sample of 65210 $\mathrm{K^{\pm}} \rightarrow \pi^{0} \pi^{0}\mathrm{e^{\pm}} \nu$ ($\mathrm{K}_{\mathrm{e}4}^{00}$) decay candidates with$1\%$ background contamination has been collected in 2003$-$2004 by the NA48/2collaboration at the CERN SPS. A study of the differential rate provides thefirst measurement of the hadronic form factor variation in the plane$(M_{\pi\pi}^2,M_{e\nu}^2)$ and brings evidence for a cusp-like structure inthe distribution of the squared $\pi^{0} \pi^{0}$ invariant mass around$4m_{\pi^+}^2$. Exploiting a model independent description of this form factor,the branching ratio, inclusive of radiative decays, is obtained using the$\mathrm{K^{\pm}} \rightarrow \pi^{0} \pi^{0} \pi^{\pm}$ decay mode asnormalization. It is measured to be BR($\mathrm{K}_{\mathrm{e}4}^{00}$) =$(2.552 \pm 0.010_{stat} \pm 0.010_{syst} \pm 0.032_{ext} ) \times 10^{-5}$,which improves the current world average precision by an order of magnitudewhile the $1.4\%$ relative precision is dominated by the external uncertaintyfrom the normalization mode. A comparison with the properties of thecorresponding mode involving a $\pi^{+} \pi^{-}$ pair($\mathrm{K}_{\mathrm{e}4}^{+-}$) is also presented.

Detailed study of the $\mathrm{K^{\pm}} \rightarrow \pi^{0} \pi^{0} \mathrm{e^{\pm}} \nu$ ($\mathrm{K}_{\mathrm{e}4}^{00}$) decay properties