Let be a positive measure deΓΏned on the product of two vector spaces E = E 1 Γ E 2 . Let F = F( ) be a natural exponential family (NEF) generated by such that the projection of F on E 1 constitutes a NEF on E 1 . This property, called a cut on E 1 , has been deΓΏned and characterized by Barndor -Nielsen (Information and Exponential Families, Wiley, Chichester) and further developed by Barndor -Nielsen and Koudou (Theory Probab. Appl. 40 (1995) 361). Their results can be used to conclude two properties of NEFs with cuts. The ΓΏrst stating that a NEF F has a cut on E 1 if and only if for all random vectors (X; Y ) on E 1 Γ E 2 , having a distribution in F, the regression curve of Y on X is linear. The second property states that the linearity of the scedastic curve of Y on X is a necessary condition for F to have a cut on E 1 . These two properties of linearity of the regression and scedastic curves provide, in some situations, rather easily veriΓΏable conditions for examining whether a NEF has a cut. Moreover, they are used to provide some interesting characterizations. In particular, some characterizations of the Gaussian and Poisson NEFs are obtained as special cases.