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A general $(N, T, \tau)$ maintenance model is developed for multicomponent systems with two types of components, namely main and auxiliary components. The main component suffers from evident failures, which are assumed to be found or detected as soon as they occur. Auxiliary components with protective or standby functions are modeled by a $k$ -out-of-$n$: $F$ subsystem, in which failures are hidden and assumed to be detected and fixed only at inspections. Although the shutdown of a subsystem may not halt the system, it could cause a potential risk to the system or financial losses. In this model, the whole system can be renewed at the $N$th failure of the main component or at time $T$, whichever occurs first. Further, incomplete periodic inspections and the optimal number of repairs before replacement are also considered in opportunistic maintenance. Incomplete periodic inspections can efficiently overcome the drawbacks of existing maintenance based on periodic inspections and opportunistic maintenance. For the case of exponential lifetime distribution, an explicit analytical expression of the maintenance cost rate in a renewal cycle is derived by applying the Laplace transform to recursive equations. By setting parameters $N,T,\tau$ to tend to infinity, respectively, special properties are derived and a comparison with several maintenance models is performed. For a given life expectancy of the system (namely $T$), the existence of optimal parameters $N$ and $\tau$ is proven. Numerical examples are presented to show the effectiveness of the proposed model. [ABSTRACT FROM AUTHOR]