We consider the light cone ('retardation') equation (LCE) of an inertially moving observer and a single worldline parameterized by arbitrary rational functions. Then a set of apparent copies, R-or C-particles, defined by the (real or complex conjugate) roots of the LCE will be detected by the observer. For any rational worldline the collective R-C dynamics is manifestly Lorentz-invariant and conservative; the latter property follows directly from the structure of Vieta formulas for the LCE roots. In particular, two Lorentz invariants, the square of total 4-momentum and total rest mass, are distinct and both integer-valued. Asymptotically, at large values of the observer's proper time, one distinguishes three types of the LCE roots and associated R-C particles, with specific locations and evolutions; each of three kinds of particles can assemble into compact large groups -clusters. Throughout the paper, we make no use of differential equations of motion, field equations, etc.: the collective R-C dynamics is purely algebraic . In both cases, at some fixed value of t, one has a whole set of roots of the considered algebraic system which determine the positions and, consequently, temporal dynamics of the collection of identical particlelike formations. In the above cited papers it was shown that for an arbitrary worldline defined by polynomial functions (except a degenerate case of zero measure) the arising collective dynamics of the system of particles-roots is necessarily conservative. This means that a complete set of conservation laws (for total momentum, angular momentum and the analogue of total energy) holds for the system of two kinds (R-or C-) of particlelike formations represented by real and complex conjugate roots of the 1 e-mail: vkassan@sci.pfu.edu.ru 2 e-mail: n.markova@mail.ru generating set of equations, respectively. It is especially interesting that all these laws follow solely from the structure of Vieta formulas for the whole system of roots, or from derivations of these formulas w.r.t. the time parameter.In the case of implicitly defined polynomial worldline [3,4] the considered algebraic dynamics is Galilei-invariant and can be compared with Newton's collective N -point dynamics. On the contrary, in the second case of a polynomial worldline implemented by the LCE of an inertially moving observer one obtains a full set of Lorentz-covariant conservation laws for the total set of R-C particles. Asymptotically, for rather great values of the observer's (proper) time T one encounters, in addition, the effects of 'self-quantization' of the admissible values of total rest mass and 'clusterization' of particle-roots. The possible meaning of the obtained algebrodynamics 3 for realistic relativistic physics requires further investigations.A detailed exposition of the above presented results can also be found in [6]. Below we generalize our consideration of polynomial dynamics to the case of a worldline parameterized by arbitrary rational functions.3 On the so-called algebrodynamical program see, e.g., [...