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Purpose: This paper aims to scrutinize the analysis of non-axisymmetric Homann stagnation point flow and heat transfer of hybrid Cu-Al2O3/water nanofluid over a stretching/shrinking flat plate. Design/methodology/approach: The similarity transformation which fulfils the continuity equation is opted to transform the coupled momentum and energy equations into the nonlinear ordinary differential equations. Numerical solutions which are elucidated in the tables and graphs are obtained using the bvp4c solver. Findings: Non-unique solutions (first and second) are feasible for both stretching and shrinking cases within the specific values of the parameters. First solution is the physical/real solution based on the execution of stability analysis. An upsurge of the ratio of the ambient fluid strain rate to the plate strain rate can delay the boundary layer separation, whereas a boost of the ratio of the ambient fluid shear rate to the plate strain rate only accelerates the separation of boundary layer. The heat transfer rate of hybrid nanofluid is greater for the stretching case than the shrinking case. However, for the shrinking case, the heat transfer rate intensifies with the increment of the copper (Cu) nanoparticles volume fraction, whereas a contrary result is found for the stretching case. Originality/value: The present numerical results are original and new. It can contribute to other researchers on electing the relevant parameters to optimize the heat transfer process in the modern industry, and the right parameters to generate non-unique solution so that no misjudgment on flow and heat transfer features. [ABSTRACT FROM AUTHOR]