  We propose the method for construction of $F(R,\xi)$ gravity model, unifying
inflation and primordial black hole formation. The proposed models are based on
the Starobinsky $R+R^2$ inflationary model, so, the function $F(R,\xi)$ is a
quadratic polynomial of the Ricci scalar $R$. We show that the potential of the
corresponding two-field chiral cosmological model in the Einstein frame can be
always found in terms of the elementary functions. The special choice of the
function $F(R,\xi)$ allows us to get such a generalization of the hybrid
inflation that can describe both inflation, and the primordial black hole
formation.
