  This article deals with the interpolating sesqui-harmonicity of a vector
field $X$ viewed as a map from a Riemannian manifold $(M,g)$ to its tangent
bundle $TM$ endowed with the Sasaki metric $g_{S}$. We show characterization
theorem for $X$ to be interpolating sesqui-harmonic map. We give also the
critical point condition which characterizes interpolating sesqui-harmonic
vector fields. When $(M,g)$ is compact and oriented and under some conditions,
we prove that $X$ is an interpolating sesqui-harmonic vector field (resp.
interpolating sesqui-harmonic map) if and only if $X$ is parallel. Moreover, we
extend this result for a left-invariant vector field on a Lie group $G$ having
a discrete subgroup $\Gamma$ such that the quotient $\Gamma\backslash G$ is
compact.
