On horizontal and complete lifts of $(1,1);$ tensor fields F satisfying the structure equation ${F(2K+S,S)=0}$

The horizontal and complete lifts from a manifold $M^{n}$ to its cotangent bundles $T^{ast }(M^{n})$ were studied by several authors. The purpose of this paper is to deal with some problems on horizontal and complete lifts tensor fields satisfying structure equation $F^{2K+S}+F^{S}=0$. We also discuss the integrability conditions and prolongations in the third tangent space $T_{3}(M^{n});$.