R. Miron and Gh. Atanasiu in [15, 16, 17] studied the geometry of $Osc^kM$. Among many various problems they solved the authors introduced the adapted basis and $d$-connection and gave its curvature theory. Different structures as almost product structure and metric structure were determined. Here, the attention is restricted onto the variational problem and integrability conditions on $E=Osc^3M$, and the transformation group is slightly different from that used in [15]. This resulted in a different theory.