Properties of solutions of two second-order differential equations with the Painlevé property

We consider a Hamiltonian system equivalent to the Painlevé II equation with respect to one component and to the Painlevé XXXIV equation with respect to another. We obtain two Bäcklund transformations (direct and inverse) of solutions of the Painlevé XXXIV equation. Based on this, we obtain a nonlinear functional relation for solutions of the Painlevé XXXIV equation with different values of its parameter. We obtain a second-degree second-order nonlinear differential equation with an arbitrary analytic function $F(t)$ and an arbitrary parameter $\gamma$ that is a Painlevé-type equation, which for $\gamma=1$ is the canonical equation XXVII in the Ince list in the case $m=2$. We obtain a Painlevé-type equation that reduces to the abovementioned equation for $F(t)=-t$ and $\gamma=0$. We show that the direct and inverse Bäcklund transformations coincide with the pair of Bäcklund transformations for the Painlevé XXXIV equation.