  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.
