Moscow International School of Physics 2022

It can be shown that an arbitrary (pseudo) Riemannian manifold can be considered as a surface in a flat space of a larger number of dimensions. This approach, for example, leads to a modified theory of gravity -- the embedding theory. However, the construction of explicit embeddings in a flat space is a nontrivial task that can be reformulated as the solution of a system of nonlinear partial differential equations. The method that will be presented in this report can be used to simplify the construction of explicit embeddings for spaces with abelian symmetry, and in some cases, to construct an explicit embedding completely. It will be shown how this method allows us to construct explicit embeddings of manifolds of (2+1) dimensional gravity: a BTZ black hole with angular momentum and a magnetic monopole.