We consider a family of linear singularly perturbed Cauchy problems which combines partial differential operators and linear fractional transforms. This work is the sequel of a study initiated in [17]. We construct a collection of holomorphic solutions on a full covering by sectors of a neighborhood of the origin in C with respect to the perturbation parameter ϵ. This set is built up through classical and special Laplace transforms along piecewise linear paths of functions which possess exponential or super exponential growth/decay on horizontal strips. A fine structure which entails two levels of Gevrey asymptotics of order 1 and so-called order 1+ is presented. Furthermore, unicity properties regarding the 1+ asymptotic layer are observed and follow from results on summability with respect to a particular strongly regular sequence recently obtained in [13] .Ministerio de Economía, Industria y Competitivida
