In this paper we study special mappings between n-dimensional (pseudo-) Riemannian manifolds. In 2003 Topalov introduced PQ ε-projectivity of Riemannian metrics, with constant ε 0, 1 + n. These mappings were studied later by Matveev and Rosemann and they found that for ε = 0 they are projective. These mappings could be generalized for case, when ε will be a function on manifold. We show that PQ ε-projective equivalence with ε is a function corresponds to a special case of F-planar mapping, studied by Mikes and Sinyukov (1983) with F = Q. Moreover, the tensor P is derived from the tensor Q and non-zero function ε. We assume that studied mappings will be also F 2-planar (Mikeš 1994). This is the reason, why we suggest to rename PQ ε mapping as F ε 2. For these mappings we find the fundamental partial differential equations in closed linear Cauchy type form and we obtain new results for initial conditions.