  We investigated both theoretically and experimentally open-ended coplanar
waveguide resonators with rf SQUIDs embedded in the central conductor at
different positions. These rf SQUIDs can be tuned by an external magnetic field
and thus may exhibit the non-centrosymmetric nonlinearity of $\chi^{(2)}$ type
with suppressed Kerr nonlinearity. We demonstrated that this nonlinearity
allows for efficient mixing of $\lambda/2$ and $\lambda$ modes in the cavity
and thus enables various parametric effects with three wave mixing. These
effects are the second harmonic generation, the half tone generation, the
parametric amplification in both degenerate and non-degenerate regimes and
deamplification in degenerate regime.
