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We consider a family of linear singularly perturbed Cauchy problems whichcombines partial differential operators and linear fractional transforms. Weconstruct a collection of holomorphic solutions on a full covering by sectorsof a neighborhood of the origin in $\mathbb{C}$ with respect to theperturbation parameter $\epsilon$. This set is built up through classical andspecial Laplace transforms along piecewise linear paths of functions whichpossess exponential or super exponential growth/decay on horizontal strips. Afine structure which entails two levels of Gevrey asymptotics of order 1 andso-called order $1^{+}$ is witnessed. Furthermore, unicity properties regardingthe $1^{+}$ asymptotic layer are observed and follow from results onsummability w.r.t a particular strongly regular sequence recently obtained in aprevious study.

On parametric Borel summability for linear singularly perturbed Cauchy problems with linear fractional transforms