The Bayesian analysis of a non-homogeneous Markov mixture of periodic autoregressions with state-dependent exogenous variables is proposed to investigate a non-linear and non-Normal time series. It is performed within a Markov chain Monte Carlo framework, along four consecutive steps: the specification of the identifiability constraint; the selection of the exogenous variables which influence the observed process and the time-varying transition probabilities of the hidden Markov chain; the choice of the cardinality of the hidden Markov chain state-space and the autoregressive order; the estimation of the parameters. The selection of the exogenous variables is performed in the complex case of correlation between variables, by means of a new procedure. An application for relating the hourly mean concentrations of sulphur dioxide with six meteorological variables, recorded for three years by an air pollution testing station located in the lagoon of Venice (Italy), is presented. The reconstruction of the sequence of the hidden states, the restoration of the missing values occurring within the observed series, the description of the periodic component are also given.