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In this article, we consider three important classes of $n$-variate fieldsindexed by the set of $N$ dimensional integers, namely stationary, stationaryincrement and self-similar fields. These classes are connected throughbijective transformations. In addition, we introduce generalized AR$(1)$ typeequations, whose unique stationary solutions are obtained via thesetransformations. Lastly, we apply the transformations in order to constructstationary fractional Ornstein-Uhlenbeck fields of the first and second kind.

One-to-one correspondences between discrete multivariate stationary, self-similar and stationary increment fields