Operator-valued semicircular elements are among the most important noncommutative random variables which are studied in operator-valued free probability theory as they are located at the crossroad of operator algebra, random matrix theory, and noncommutative algebra. These noncommutative random variables enjoy the important feature that their operator-valued Cauchy transforms are completely determined by the so-called Dyson equation, which is a kind of quadratic equation depending only on their mean and covariance map. While the covariance maps of operator-valued semicircular elements are necessarily completely positive, the Dyson equation itself can be considered without this restriction. In my talk, I will explain that already 2-positivity is enough to ensure various analytic properties of their solutions, including an operator-valued version of the inviscid Burgers equation. In doing so, we will see that tools from noncommutative function theory serve as a substitute for the lacking realization by operator-valued semicircular elements.