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The Resonant Switch (RS) model of twin high-frequency quasi-periodicoscillations (HF QPOs) observed in neutron star binary systems, based on switchof the twin oscillations at a resonant point, has been applied to the atollsource 4U 1636-53 under assumption that the neutron star exterior can beapproximated by the Kerr geometry. Strong restrictions of the neutron starparameters M (mass) and a (spin) arise due to fitting the frequency pairsadmitted by the RS model to the observed data in the regions related to theresonant points. The most precise variants of the RS model are those combiningthe relativistic precession frequency relations with their modifications. Here,the neutron star mass and spin estimates given by the RS model are confrontedwith a variety of equations of state (EoS) governing structure of neutron starsin the framework of the Hartle-Thorne theory of rotating neutron stars appliedfor the observationally given rotation frequency f_rot~580 Hz (or alternativelyf_rot~290 Hz) of the neutron star at 4U 1636-53. It is shown that only twovariants of the RS model based on the Kerr approximation are compatible withtwo EoS applied in the Hartle-Thorne theory for f_rot~580 Hz, while no variantof the RS model is compatible for f_rot~290 Hz. The two compatible variants ofthe RS model are those giving the best fits of the observational data. However,a self-consistency test by fitting the observational data to the RS model withoscillation frequencies governed by the Hartle-Thorne geometry described bythree spacetime parameters M, a and (quadrupole moment) q related by the twoavailable EoS puts strong restrictions. The test admits only one variant of theRS model of twin HF QPOs for the Hartle-Thorne theory with the Gandolfi et al.(2010) EoS predicting the parameters of the neutron star $M \sim 2.10\mathrm{M}_{\odot}$, $a \sim 0.208$, and $q/a^2 \sim 1.77$.

Equations of state in the Hartle-Thorne model of neutron stars selecting  acceptable variants of the resonant switch model of twin HF QPOs in the atoll  source 4U 1636-53