  This study investigates the geometric linearization of constraint Hamiltonian
systems using the Jacobi metric and the Eisenhart lift. We establish a
connection between linearization and maximally symmetric spacetimes, focusing
on the Noether symmetries admitted by the constraint Hamiltonian systems.
Specifically, for systems derived from the singular Lagrangian $$ L\left(
N,q^{k},\dot{q}^{k}\right) =\frac{1}{2N}g_{ij}\dot{q}^{i}\dot{q}^{j}-NV(q^{k}),
$$ where $N$ and $q^{i}$ are dependent variables and $\dim g_{ij}=n$, the
existence of $\frac{n\left( n+1\right) }{2}$ Noether symmetries is shown to be
equivalent to the linearization of the equations of motion. The application of
these results is demonstrated through various examples of special interest.
This approach opens new directions in the study of differential equation
linearization.
