This paper is predominantly devoted to thoroughgoing investigation of the Legendre foliated cocycles which are totally compatible with second order ordinary differential equation fields (SODE). For this goal, the spencer theory of formal integrability, relative 1-forms, Bott connection and the Vrăceanu connection, are applied as the prominent tools in order to designate the sufficient conditions for the metric concomitant to the analyzed SODE structure, to extend to a bundle-like metric for the lifted Legendre foliated cocycle on the tangent space of an arbitrary contact manifold. Significantly, the geometric structure of the constructed metric foliated cocycles on the tangent bundle are extensively scrutinized by virtue of the two local invariants corresponding to Legendre foliations comprising a symmetric 2-form and a symmetric 3-form characterizing on the tangent bundle of the foliation. In particular, it is focused on the essence of the metric Legendre foliated cocycles adapted to SODE structure on two specific class of contact metric manifolds including the Sasakian manifolds and
(
k
,
μ
)
(k,\mu )
-spaces. Furthermore, the pivotal circumstances regarding the existence and the geometric features of the conjugate Legendre foliated cocycles on the tangent space of the contact metric manifolds which are entirely compatible with the ambient SODE structure are conclusively explored.