  We consider a class of first-order partial differential operators, acting on
the space of ultradifferentiable periodic functions, and we describe their
range by using the following conditions on the coefficients of the operators:
the connectedness of certain sublevel sets, the dimension of the subspace
generated by the imaginary part of the coefficients, and Diophantine
conditions. In addition, we show that these properties are also linked to the
regularity of the solutions. The results extend previous ones in Gevrey
classes.
