  In the Standard Model (SM) the branching ratio for the decay $K_L \to \nu
\bar{\nu}$ into two neutrinos is helicity suppressed and predicted to be very
small $ \leq O(10^{-17})$. We consider two natural extensions of the SM, such
as two-Higgs-doublet model (2HDM) and the $\nu$MSM with additional singlet
scalar, those main features is that they can lead to an enhanced $Br(K_L \to
invisible)$. In the 2HDM the smallness of the neutrino mass is explained due to
the smallness of the second Higgs doublet vacuum expectation value. Moreover,
the $\nu$MSM extension with additional singlet field can explain the $(g - 2)$
muon anomaly. The considered models demonstrate that the $K_L \to invisible $
decay is a clean probe of new physics scales well above 100 TeV, that is
complementary to rare $K \to \pi+invisible$ decay, and provide a strong
motivation for its sensitive search in a near future low-energy experiment.
