  Studies on double beta decay processes in $^{106}$Cd were performed by using
a cadmium tungstate scintillator enriched in $^{106}$Cd at 66%
($^{106}$CdWO$_4$) with two CdWO$_4$ scintillation counters (with natural Cd
composition). No effect was observed in the data accumulated over 26033 h. New
improved half-life limits were set on the different channels and modes of the
$^{106}$Cd double beta decay at level of $\lim T_{1/2}\sim 10^{20}-10^{22}$ yr.
The limit for the two neutrino electron capture with positron emission in
$^{106}$Cd to the ground state of $^{106}$Pd,
$T^{2\nu\mathrm{EC}\beta^+}_{1/2}\geq2.1\times 10^{21}$ yr, was set by the
analysis of the $^{106}$CdWO$_4$ data in coincidence with the energy release
511 keV in both CdWO$_4$ counters. The sensitivity approaches the theoretical
predictions for the decay half-life that are in the range
$T_{1/2}\sim10^{21}-10^{22}$ yr. The resonant neutrinoless double-electron
capture to the 2718 keV excited state of $^{106}$Pd is restricted at the level
of $T^{0\nu\mathrm{2K}}_{1/2}\geq2.9\times10^{21}$ yr
