Abstract. We extend the concept of  constructible differentially finite algebraic (CDF) series (introduced by C.~Reutenauer and the first author) to the context of several variables, and show that the series solution of first order differential equations  y'=x(t,y)  and functional equation  y=x(t,y), with x  CDF in two variables, are CDF-series. We also give many effective closure properties for CDF-series in several variables. CDF-series appear naturally in the study of enumeration problems in a manner similar to D-finite series which have been shown to have  great importance in
enumerative combinatorics by Gessel, Stanley  and  Zeilberger.