We present the construction of distinguished non-Kähler metrics on non-compact Calabi-Yau 3-folds. These metrics solve a system of equations known as the IIB system which arises in theoretical physics and is related to recent attempts to define notions of "canonical" metrics on non-Kähler Calabi-Yau manifolds. The examples we construct include infinitely many complete metrics obtained by deforming an asymptotically conical Calabi-Yau 3-folds in the direction of a non-trivial Äppli class and families of solutions on the ordinary double point and its smoothing that enjoy a cohomogeneity one symmetry (i.e. there is a symmetry group that acts with 1-dimensional orbit space) and produce a "conifold transition" in non-Kähler Calabi-Yau geometry. The talk is based on joint work with Mario Garcia-Fernandez.