In relation to the solution of the Vekua system for axial type monogenic functions, generalizations of Fueter’s Theorem are discussed. We show that if f is a holomorphic function in one complex variable, then for any underlying space Rⁿ₁ the induced function ∆^k+(n−1)/2f(x₀ +x)P_k(x), where P_k(x) is left-monogenic and homogeneous of degree k, is left-monogenic whenever k+(n−1)/2 is a non-negative integer. If the space dimension n + 1 is odd, then the above also holds for k being non-negative integers.