  We provide sufficient conditions of P\'olya type which guarantee the positive
definiteness of a $2\times 2$-matrix-valued function in $\mathbb{R}$ and
$\mathbb{R}^3$. Several bivariate covariance models have been proposed in
literature, where all components of the covariance matrix are of the same
parametric family, such as the bivariate Mat\'{e}rn model. Based on the P\'olya
type conditions, we introduce two novel bivariate parametric covariance models
of this class, the powered exponential (or stable) covariance model and the
generalized Cauchy covariance model. Both models allow for flexible smoothness,
variance, scale, and cross-correlation parameters. The smoothness parameters
are in $(0, 1]$. Additionally, the bivariate generalized Cauchy model allows
for distinct long range parameters. We also show that the univariate spherical
model can be generalized to the bivariate case within the above class only in a
trivial way. In a data example on the content of copper and zinc in the top
soil of Swiss Jura we compare the bivariate powered exponential model to the
traditional linear model of coregionalization and the bivariate Mat\'{e}rn
model.
