= Solution
The spatial coordinates are cyclic because the flat <Friedmann-Lemaître-Robertson-Walker metric> depends only on time. Differentiating the printed affine Lagrangian gives the <canonical momentum>
$$
P_i^c=-m g_{i\nu}\dot x^\nu=-ma^2\dot x^i.
$$
The <Euler-Lagrange equations> give $dP_i^c/d\tau=\partial L/\partial x^i=0$, so
$$
\boxed{\mathbf P^c=\text{constant}.}
$$
This is the conserved quantity associated with spatial <translation invariance>. The overall minus sign in the stated action makes this canonical covector opposite to the usual future-directed mechanical momentum; it does not alter the geodesic trajectories or the conservation result. The <canonical momentum sign in the affine particle action> will be kept explicit below.
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