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Synchronously pumped optical parametric oscillators (OPOs) are highly efficient sources of long-wavelength pulses and nonclassical light, making them invaluable for applications in spectroscopy, metrology, multi-photon microscopy, and quantum computation. Typical systems based on free-space cavities either operate non-degenerately, which limits their efficiency, or use active feedback control to achieve degenerate operation, which limits these systems to dedicated low-noise environments. In this work, we demonstrate a femtosecond monolithically integrated OPO. In contrast with bulk OPOs, our monolithic 10 GHz cavity, based on reverse-proton-exchanged lithium niobate, operates stably without active locking. By detuning the repetition rate of the free-running pump laser from the cavity free spectral range, we control the intracavity pulse dynamics and observe many of the operating regimes previously encountered in free-space degenerate OPOs, such as box-pulsing and quadratic bright-dark solitons (simultons), in addition to non-degenerate operation. When operated in the simulton regime and pumped with 125 fs pulses at 1 µm, this monolithic OPO chip outputs broadband sech²pulses (63 nm, 3 dB) with tens of milliwatts of average power.