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An experiment to study double-beta decay processes in $^{106}$Cd using a $^{106}$CdWO$_4$ crystal scintillator (mass 215.4 g) enriched in $^{106}$Cd to 66$\%$ is in progress in the DAMA/R&D setup at LNGS. The enriched crystal was placed in a close geometry with two CdWO$_4$ crystal scintillators in order to increase the detection efficiency to $\gamma$'s that can be emitted in the double-beta decay processes in $^{106}$Cd. The data has been accumulated for 467 days considering the coincidence and/or anticoincidence events in the three detectors. No effects have been observed and only lower limits on the half-lives for the double-beta decay processes in $^{106}$Cd have been set. They are at the level of lim T$_{1/2}$$\sim$10$^{20}$-10$^{22}$ years. The limit on the half-life for the $2\nu\varepsilon\beta^+$ in $^{106}$Cd was preliminarily estimated as T$_{1/2}$ $\ge$2.1$\times$10$^{21}$ yr, which approaches the theoretical expectations for this process that are in the range of T$_{1/2}$ = 10$^{21}$-10$^{22}$ yr. Such results could contribute in principle to the estimation of the effective nuclear matrix elements for $2\beta$ decay processes, which can be considered one of the most challenging theoretical problem that hinders precision studies of $0\nu2\beta$ decay in the case of the event of a discovery.
