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A conjecture is formulated about the connection between isometries of vacuum space times and the existence of corresponding space times with electromagnetic fields for which the structure of electromagnetic four potentials is readily predicted. One example is presented showing how this conjecture works, three others are only commented.

It is shown that some simple and interesting analogy between the plane and the spherical solutions of Einstein gravitational equations in vacuum exists. This analogy is applied to the solutions of some problems of motions in the plane gravitational field.

The equations of motion for the plane shell consisting of an ideal fluid and creating the de Sitter solution on one side and the static plane symmetric solution of Einstein equations with positive cosmological constant on the other side are derived. Solution of the equation of motion for the case of “plane gravitational condensor” is found and the necessary agreement with the solution of the problem in the caseA = 0 is demonstrated.

The geometric interpretation of Zelmanov's Chronometric invarians is given and their conection to observable quantities as well as the connection of Zelmanov's approach to the generally covariant Dehnen's method is found.

From the requirement of the plane symmetry the form of the exterior and static gravitational field of the plane shell is found. Its geometry is demonstrated by embedding the selected sections into the plane space. The exterior solution is linked up with the plane and with the couple of plane-parallel planes at rest. It is shown that the existence of strong pressures in the planes (which are necessary for the manifestation of the rest of both planes) makes it difficult to compare it directly with the corresponding classic case. If the pressure is relaxed, the planes will move towards one another by hyperbolic motion. It is shown that the hyperbolic acceleration of the plane is the same as in the classical case. The gravitational field outside the plane remains static so that no gravitational radiation arises.

The Rayski method is used to calculate the energy of a static and spherically symmetrical gravitational system.

In recent years an increased interest in General Relativity Theory (GRT) has been apparent among theoretical physicists, mathematicians, astronomers and geophysicists. At the same time, the necessity of rinding a final and generally satisfactory solution of a very important problem, namely that of conservation laws, has become topical. Surprising difficulties have been encountered in the attempt to transfer these concepts to General Relativity Theory. In the course of a few years a comprehensive literature has resulted and now the time is ripe for a thorough summary of the results attained so far.

The method of a tetrad field proposed by Rayski was used to determine the energy-momentum tensor for an approximate solution of the Einstein equations in the static case, from which it follows that the static gravitational field has non-zero mass but zero momentum, as was to be expected.