Summary: Pseudodifferential equations of the form \(v(D_{\chi})y=f,\) where \(v\) is a function holomorphic at zero and \(D_{\chi}\) is a pseudodifferential operator, are studied on spaces of test functions of non-Gaussian infinite-dimensional analysis. The results obtained are applied to construct a generalized translation operator \(T_{y}^{\chi}=\chi(\langle y,D_{\chi}\rangle)\) on the mentioned spaces and to study its properties. In particular, the associativity, the commutativity, and another properties of \(T_{y}^{\chi}\) which are analogs of the classical properties of the generalized translation operator are investigated.