Vector fields and flows on $C^{\mathrm{\infty }}$-schemes

We prove that a vector field on an affine $C^{\mathrm{\infty }}$-scheme Spec(A) has a flow if the $C^{\mathrm{\infty }}$-ring A is finitely generated. If the vector field is complete then the flow is the target map of a groupoid internal to the category of $C^{\mathrm{\infty }}$-schemes.