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The work characterizes when is the equation $y^{\left(n\right)}+\mu p\left(x\right)y=0$ eventually disconjugate for every value of $\mu$ and gives an explicit necessary and sufficient integral criterion for it. For suitable integers $q$, the eventually disconjugate (and disfocal) equation has 2-dimensional subspaces of solutions $y$ such that $y^{\left(i\right)}>0$, $i=0,...,q-1$, ${(-1)}^{i-q}{y}^{\left(i\right)}>0$, $i=q,...,n$. We characterize the “smallest” of such solutions and conjecture the shape of the “largest” one. Examples demonstrate that the estimates are sharp.

We discuss an open question of Kiguradze and Chanturia about Property $A$ and Property $B$ for the equation $y^{\left(n\right)}+py=0$. The proposed integral criterion is proved in a few cases.