Schwarzschild Massive-Point-Particle Problem in Arbitrary Radial Gauge

We present, for the first time, a correct solution of the Schwarzschild Massive Point (SMP) problem in {\em arbitrary} radial gauge and formulate the strict mathematical assumptions, which are necessary and sufficient for this. In GR, there exists a two-parameter family of such exact SMP solutions to the Einstein equations, which are physically distinguishable from the well-studied one parameter family of vacuum Schwarzschild solutions related with Schwarzschild Black Holes (SBH). The obtained here SMP family of solutions is defined by positive bare mass $M_0>0$ and positive Kepler mass $M<M_0$, or, alternatively, by the standard gravitational radius $\rho_g$ and mass ratio $\varrho=M/M_0 \in (0,1)$. The metrics of spacetime have an unavoidable jump at the place of a massive point particle. We also present a proper development of the theory of distribution defined by kernels with a finite jump which appear in the solution of SMP. The specific properties of these distributions are used for work with SMP. A series of physical properties of SMP solutions are derived and commented. Our findings are important for description of Extremely Compact Objects (ECOs) studied in relation with possible echoes in Gravitational Waves (GW) recently discovered by the LIGO/VIRGO collaboration.