Analysis of the equations of motion of fictitious matter in embedding theory

Embedding theory is a possible modification of general relativity that provides a framework for explaining the observed effects typically attributed to dark matter. The idea of this modification is to consider our spacetime as a four-dimensional surface in a ten-dimensional flat ambient space. The equations of motion in embedding theory can be reformulated as a set of Einstein equations with the contribution of some additional fictitious matter and of equations describing this matter. We analyze static solutions of these equations, which are reduced to fictitious-matter configurations of the wall, string, and ball types. The string case is ultimately described by the Liouville equation, and the ball case is described by its three dimensional analogue. For the string case, we show that as the density contribution decreases at infinity, all solutions to the Liouville equation are rotationally symmetric. For the case of the ball, we show that under the assumption of spherical symmetry, there exists a unique one-parameter family of solutions that are smooth at the center.