In this paper it is shown that the optimal design for estimating any specific parameter in a polynomial spline regression with a single multiple knot at the center is supported by one of two sets of points. Familiarity with the notation and terminology used in the paper by Studden (1968) is assumed.
"Optimal Designs with a Polynomial Spline Regression with a Single Multiple Knot at the Center." Ann. Math. Statist. 42 (3) 952 - 960, June, 1971. https://doi.org/10.1214/aoms/1177693324