With a limited budget like that you will not be able to find a new laptop with anything but integrated graphics. You are basically limited to a laptop with a dual core Core i5-4210u that has the integrated Intel HD 4400 graphics core, or a quad core AMD A10-5750m that has the Radeon HD 8650g integrated graphics core. You will get better performance with the A10-5750m in games.
You can by the ThinkPad E545 with a starting price of $499 and upgrade the APU to the A10-5750m for another $70; $579 in total before tax. However, it only comes with one stick of 4GB RAM. You will need to install another stick of 4GB of RAM to get the best results because with only 1 stick of RAM the data transfer rate only runs at half speed. Two sticks will allow RAM to operate at full speed.
Because the laptop will rely on system RAM for graphics, RAM running at half speed will cause game performance to be lower. You can upgrade the RAM at Lenovo's website to 8GB for another $80. Or you can buy a stick of 4GB RAM yourself for about $40 - $50 and install it yourself later.
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/shop.lenovo.com/us/en/laptops/thinkpad/e-series/e545/
Closer to the $500 mark would be the Lenovo G50. You can get the $529 or $539 version. The only difference is 6GB vs 8GB and a standard HDD vs a Hybrid HDD. Both come with the i5-4210u which has a base speed of 1.7GHz but with Turbo Boost it can get up to 2.7GHz when one core is used or 2.5GHz when both cores are used.
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/shop.lenovo.com/us/en/laptops/lenovo/g-series/g50-70/
The less expensive version for $449 only comes with an i3-4030u CPU running at 1.9GHz and 4GB of RAM. Buying and installing 4GB of RAM yourself bumps that price up by another $40 - $50.