  Triple beta decay of $^{150}$Nd to the ground and excited states of
$^{150}$Eu and quadruple beta decay of $^{150}$Nd to the excited states of
$^{150}$Gd have been studied using a 400 cm$^3$ low-background HPGe detector
and an external source consisting of 3046 g of natural Nd$_2$O$_3$ powder. A
half-life limit for the quadruple beta decay to the 0$_1^+$ state of $^{150}$Gd
was found to be $T_{1/2} (0\nu+4\nu) > 8.7\times10^{20}$ yr (90\% C.L.). For
other (0$\nu$+4$\nu$) transitions to the 2$_1^+$, 3$_1^-$, 4$^+_1$, 2$_2^+$ and
2$^+_3$ excited states limits for the first time have been obtained at the
level of $(6.1-9.5)\times10^{20}$ yr (90\% C.L.). We report here also the
results for the first search for triple beta decay to the ground and excited
final states of $^{150}$Eu. We find no evidence of this decay and set lower
limits on the half-life in the range $T_{1/2} (0\nu+3\nu) >
(0.04-4.8)\times10^{20}$ yr (90\% C.L.).
