The NA62 experiment at CERN reports searches for

$K^+ \to e^+ N$, $K^+ \to \mu^+ N$ and $K^+ \to \mu^+ X$ decays,

where $N$ and $X$ are massive invisible particles, using the 2016-2018 data set.

The $N$ particle is assumed to be a heavy neutral lepton, and the results are expressed as upper limits

of $O(10^{-9})$ and $O(10^{-8})$ on the neutrino mixing parameter $|U_{e4}|^2$ and $|U_{\mu 4}|^2$,

improving on the earlier searches for heavy neutral lepton production and decays in the kinematically accessible mass range.

The $X$ particle is considered a scalar or vector hidden sector mediator decaying to an invisible final

state, and upper limits of the decay branching fraction for $X$ masses in the range 10-370 MeV/c$^2$ are

reported for the first time, ranging from $O(10^{-5})$ to $O(10^{-7})$.

An upper limit of

$1.0 \cdot 10^{-6}$

is established

at 90\% CL on the

$K^+\to \mu^+ \nu \nu \bar{\nu}$ branching fraction.