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Space missions have discovered a large number of exoplanets evolving in (or close to) mean-motion resonances (MMRs) and resonant chains. Often, the published data exhibit very high uncertainties due to the observational limitations that introduce chaos into the evolution of the system on especially shorter or longer timescales. We propose a study of the dynamics of such systems by exploring particular regions in phase space. We exemplify our method by studying the long-term orbital stability of the three-planet system Kepler-51 and either favor or constrain its data. It is a dual process which breaks down in two steps: the computation of the families of periodic orbits in the 1:2:3 resonant chain and the visualization of the phase space through maps of dynamical stability. We present novel results for the general four-body problem. Stable periodic orbits were found only in the low-eccentricity regime. We demonstrate three possible scenarios safeguarding Kepler-51, each followed by constraints. Firstly, there are the 2/1 and 3/2 two-body MMRs, in which $e_b<0.02$, such that these two-body MMRs last for extended time spans. Secondly, there is the 1:2:3 three-body Laplace-like resonance, in which $e_c<0.016$ and $e_d<0.006$ are necessary for such a chain to be viable. Thirdly, there is the combination comprising the 1/1 secondary resonance inside the 2/1 MMR for the inner pair of planets and an apsidal difference oscillation for the outer pair of planets in which the observational eccentricities, $e_b$ and $e_c$, are favored as long as $e_d\approx 0$. With the aim to obtain an optimum deduction of the orbital elements, this study showcases the need for dynamical analyses based on periodic orbits performed in parallel to the fitting processes.