As explained in the first volume (Honda et al. Virtual Turning Points. Springer Briefs in Mathematical Physics, vol. 4. Springer-Verlag, Berlin, 2015), the virtual turning points are defined in terms of the bicharacteristic strips of (the Borel transform of) a system of differential equations. However, when we consider its tangential system, a new kind of turning points called non-hereditary turning points appears. Using a system with an integral representation of solutions mainly, we discuss non-hereditary turning points in this chapter. After reviewing the definition of a virtual turning point and an s-virtual turning point, its special version for a system with integral representations, we explain the non-hereditary turning points and their fundamental properties. As an example, we discuss a tangential system of the Pearcey system to a hyperplane in detail. We also present some intriguing examples of the Stokes geometry of systems with integral representation of solutions.