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A new branch of black hole solutions occurs along with the standardSchwarzschild branch in $n$-dimensional extensions of general relativityincluding terms quadratic in the Ricci tensor. The standard and new branchescross at a point determined by a static negative-eigenvalue eigenfunction ofthe Lichnerowicz operator, analogous to the Gross-Perry-Yaffe eigenfunction forthe Schwarzschild solution in standard $n=4$ dimensional general relativity.This static eigenfunction has two r\^oles: both as a perturbation away fromSchwarzschild along the new black-hole branch and also as a threshold unstablemode lying at the edge of a domain of Gregory-Laflamme-type instability of theSchwarzschild solution for small-radius black holes. A thermodynamic analogywith the Gubser and Mitra conjecture on the relation between quantumthermodynamic and classical dynamical instabilities leads to a suggestion thatthere may be a switch of stability properties between the old and newblack-hole branches for small black holes with radii below the branch crossingpoint.

Lichnerowicz Modes and Black Hole Families in Ricci Quadratic Gravity