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We report the experimental studies of a parametric excitation of a second sound (SS) by a first sound (FS) in a superfluid helium in a resonance cavity. The results on several topics in this system are presented: (i) The linear properties of the instability, namely, the threshold, its temperature and geometrical dependencies, and the spectra of SS just above the onset were measured. They were found to be in a good quantitative agreement with the theory. (ii) It was shown that the mechanism of SS amplitude saturation is due to the nonlinear attenuation of SS via three wave interactions between the SS waves. Strong low frequency amplitude fluctuations of SS above the threshold were observed. The spectra of these fluctuations had a universal shape with exponentially decaying tails. Furthermore, the spectral width grew continuously with the FS amplitude. The role of three and four wave interactions are discussed with respect to the nonlinear SS behavior. The first evidence of Gaussian statistics of the wave amplitudes for the parametrically generated wave ensemble was obtained. (iii) The experiments on simultaneous pumping of the FS and independent SS waves revealed new effects. Below the instability threshold, the SS phase conjugation as a result of three-wave interactions between the FS and SS waves was observed. Above the threshold two new effects were found: a giant amplification of the SS wave intensity and strong resonance oscillations of the SS wave amplitude as a function of the FS amplitude. Qualitative explanations of these effects are suggested.