  In development of the old conjecture of Stuckelberg, Wheeler and Feynman on
the so-called "one electron Universe", we elaborate a purely algebraic
construction of an ensemble of identical pointlike particles occupying the same
worldline and moving in concordance with each other. In the proposed
construction one does not make use of any differential equations of motion,
Lagrangians, etc. Instead, we define a "unique" worldline implicitly, by a
system of nonlinear polynomial equations containing a time-like parameter. Then
at each instant there is a whole set of solutions defining the coordinates of
particles-copies localized on the unique worldline and moving along it. There
naturally arise two different kinds of such particles which correspond to real
or complex conjugate roots of the initial system of polynomial equations,
respectively. At some particular time instants, one encounters the transitions
between these two kinds of particles-roots that model the processes of
annihilation or creation of a pair "particle-antiparticle". We restrict by
consideration of nonrelativistic collective dynamics of the ensemble of such
particles on a plane. Making use of the techniques of resultants of
polynomials, the generating system reduces to a pair of polynomial equations
for one unknown, with coefficients depending on time. Then the well-known Vieta
formulas predetermine the existence of time-independent constraints on the
positions of particles-roots and their time derivatives. We demonstrate that
for a very wide class of the initial polynomials (with polynomial dependence of
the coefficients on time) these constraints always take place and have the form
of the conservation laws for total momentum, angular momentum and (the analogue
of) total mechanical energy of the "closed" system of particles.
