| Let X be a nonsingular subvariety in projective space covered by lines. For a point x in X, denote by C_x the subvariety of the projective tangent space of X at x, corresponding to the set of lines through x lying on X. When the family C_x is isotrivial as x varies over general points of X, we say that X has isotrivial VMRT-structure. We will discuss the question `is a Fano manifold of Picard number 1 with isotrivial VMRT-structure quasi-homogeneous?' We will give an affirmative answer when C_x satisfies certain conditions, which hold for complete intersections of dimension bigger than 1 and of multi-degree different from (2,2), (2,3), (2,2,2). This implies that a Fano complete intersection of index bigger than 3 and multi-degree different from (2), (3), (2,2), (2,2,2) cannot have isotrivial VMRT-structure. |