Near-infrared spectroscopy (NIRS) has recently been investigated as a non-invasive brain-computer interface (BCI) for individuals with severe motor impairments. For the most part, previous research has investigated the development of NIRS-BCIs operating under synchronous control paradigms, which require the user to exert conscious control over their mental activity whenever the system is vigilant. Though functional, this is mentally demanding and an unnatural way to communicate. An attractive alternative to the synchronous control paradigm is system-paced control, in which users are required to consciously modify their brain activity only when they wish to affect the BCI output, and can remain in a more natural, 'no-control' state at all other times. In this study, we investigated the feasibility of a system-paced NIRS-BCI with one intentional control (IC) state corresponding to the performance of either mental arithmetic or mental singing. In particular, this involved determining if these tasks could be distinguished, individually, from the unconstrained 'no-control' state. Deploying a dual-wavelength frequency domain near-infrared spectrometer, we interrogated nine sites around the frontopolar locations (International 10-20 System) while eight able-bodied adults performed mental arithmetic and mental singing to answer multiple-choice questions within a system-paced paradigm. With a linear classifier trained on a six-dimensional feature set, an overall classification accuracy of 71.2% across participants was achieved for the mental arithmetic versus no-control classification problem. While the mental singing versus no-control classification was less successful across participants (62.7% on average), four participants did attain accuracies well in excess of chance, three of which were above 70%. Analyses were performed offline. Collectively, these results are encouraging, and demonstrate the potential of a system-paced NIRS-BCI with one IC state corresponding to either mental arithmetic or mental singing.