  Flavor SU(3) symmetry, including $30\%$ first order SU(3) breaking, has been
shown to describe adequately a vast amount of data for charmed meson decays to
two pseudoscalar mesons and to a vector and a pseudoscalar meson. We review a
recent dramatic progress achieved by applying a high order perturbation
expansion in flavor SU(3) breaking and treating carefully isospin breaking. We
identify a class of U-spin related $D^0$ decays to pairs involving charged
pseudoscalar or vector mesons, for which high-precision nonlinear amplitude
relations are predicted. Symmetry breaking terms affecting these relations are
fourth order U-spin breaking, and terms which are first order in isospin
breaking and second order in U-spin breaking. The predicted relations are shown
to hold experimentally at a precision varying between $10^{-3}$ and $10^{-4}$,
in agreement with estimates of high order terms. We also discuss amplitude
relations for $D^0$ decays to pairs of neutral pseudoscalar mesons, and
relations for rate asymmetries between decays involving $K^0_S$ and $K^0_L$
which hold up to second order U-spin breaking.
