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We present the photometric determination of the bright-end (L_Lya>10^43.5 erg/s) of the Lya luminosity function (LF) within four redshifts windows in the interval 2.2<z<3.3. Our work is based on the Javalambre Photometric Local Universe Survey (J-PLUS) first data-release, which provides multiple narrow-band measurements over ~1000 deg^2, with limiting magnitude r~22. The analysis of high-z Lya-emitting sources over such a wide area is unprecedented, and allows to select a total of ~14,500 hyper-bright (L_Lya>10^43.3 erg/s) Lya-emitting candidates. We test our selection with two spectroscopic follow-up programs at the GTC telescope, confirming ~89% of the targets as line-emitting sources, with ~64% being genuine z~2.2 QSOs. We extend the 2.2<z<3.3 Lya LF for the first time above L_Lya~10^44 erg/s and down to densities of ~10^-8 Mpc^-3. Our results unveil with high detail the Schechter exponential-decay of the brightest-end of the Lya LF, complementing the power-law component of previous LF determinations at 43.3<Log_10(L_Lya / [erg/s])<44. We measure Phi^*=(3.33+-0.19)x10^-6, Log(L^*)=44.65+-0.65 and alpha=-1.35+-0.84 as an average over the redshifts we probe. These values are significantly different than the typical Schechter parameters measured for the Lya LF of high-z star-forming LAEs. This suggests that z>2 AGN/QSOs (likely dominant in our samples) are described by a structurally different LF than z>2 star-forming LAEs, namely with L^*_QSOs ~ 100 L^*_LAEs and Phi^*_QSOs ~ 10^-3 Phi^*_LAEs. Finally, our method identifies very efficiently as high-z line-emitters sources without previous spectroscopic confirmation, currently classified as stars (~2000 objects in each redshift bin, on average). Assuming a large predominance of Lya-emitting AGN/QSOs in our samples, this supports the scenario by which these are the most abundant class of z>2 Lya emitters at L_Lya>10^43.3 erg/s.