  Lines of exceptional points are robust in the 3-dimensional non-Hermitian
parameter space without requiring any symmetry. However, when more elaborate
exceptional structures are considered, the role of symmetry becomes critical.
One such case is the exceptional chain (EC), which is formed by the
intersection or osculation of multiple exceptional lines (ELs). In this study,
we investigate a non-Hermitian classical mechanical system and reveal that a
symmetry intrinsic to second-order dynamical equations, in combination with the
source-free principle of ELs, guarantees the emergence of ECs. This symmetry
can be understood as a non-Hermitian generalized latent symmetry, which is
absent in prevailing formalisms rooted in first-order Schr\"odinger-like
equations and has largely been overlooked so far. We experimentally confirm and
characterize the ECs using an active mechanical oscillator system. Moreover, by
measuring eigenvalue braiding around the ELs meeting at a chain point, we
demonstrate the source-free principle of directed ELs that underlies the
mechanism for EC formation. Our work not only enriches the diversity of
non-Hermitian degeneracies, but also highlights the new potential for
non-Hermitian physics in second-order dynamical systems.
